#discrete random variables
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mathmedicine · 1 year ago
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AS LEVEL STATISTICS 1 (PAPER 5)
STATISTICS 1 (2002-2010) C1 Representation of Data C2 Permutations and Combinations C3 Probability C4 Discrete Random Variables C5 The Normal Distribution
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three-blogs-in-a-trenchcoat · 1 year ago
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If you don't mind becoming a girl (and assuming that once you've "became a girl" - whatever that means - you can't push the button anymore, but until then you can push it an unbounded amount of times), then you can just keep pushing the button, and expect to win $99M before being femmed.
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samkerrworshipper · 10 months ago
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underneath the surface pt.2
i wrote a part two to a fic? sorry what?
anyways loved will always have a soft spot for this! hope you all enjoy and lmk how you felt about it xo
warnings: soft smut & soft angst viewer discretion advised
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Alexia knows she should be the brave face in this whole situation, she’s not the one sitting in a hospital gown on a gurney about to head into surgery, but that doesn’t change the fact that she’s sizzling with anxiety. She doesn’t know why, there isn’t anything particularly terrifying about her circumstances, this is a routine surgery, the chance of complication is extremely low, the chance of death is less than 1%. She knows because she’s google checked five times in the last hour.
She’s trying to hide it, because she knows even though you are putting on a brave face you are secretly terrified. She’s been playing the chivalrous girlfriend. Trying to eliminate any outside stress so that you don’t have to worry about it, which is causing Alexia to stumble into a spiral of worries. She’s worried for you, she’s worried for the future, she’s worried about everything that moves and breathes.
Alexia can normally cope under stress, give her a game winning penalty in a world cup winning game and she’ll treat it with the same amount of composure as she would a penalty in a 20-0 friendly game.
Alexia stressing about you though, is a completely different story. She’s snapped at everyone in the lead up to this week, she’s shut down, she’s in her fortress and until she knows that you’re going to be okay that’s not going to change.
You’re the best thing that happened to her, and even though she has absolutely zero impact on what’s going to happen in the operation room she feels like if she controls all of the outside variables then somehow that’s her contribution. She wants to believe that if she can make everything on the outside right, then it’ll somehow change what’s happening on the inside, Alexia wishes deep down that she could fix it all and that this situation wasn’t happening.
Alexia doesn’t understand why you suffer so much from a disease that has no cure, she doesn’t understand why something so horrible exists and why you are plagued with it.
You don’t seem phased at all, like this is normal routine for you which is making Alexia feel even worse.
You’ve gone about your morning as usual, denying Alexia in every way as she tried to do random acts of service that were actually useless.
You’d quite literally breezed your way through it all and to the hospital without a waiver, no tears, no worries, no stress. It was a long time coming, Alexia appreciated the fact that from what you and google had told her, laparoscopy’s could be life changing for people with endometriosis, it could effectively make your quality of life during your period so much better. But it was still surgery, Alexia remembers when she had her acl surgery, she was a bundle of nerves to the point where her mami had to sit by her bed to stop her from hyperventilating whilst they administered the anaesthesia.
There is no clutching for Alexia’s hand, no desperate admissions of fear, no articulation of any anxieties or stress. You’re practically meditative, sitting on the hospital bed in your gown waiting patiently for the doctors to come and collect you.
Alexia’s walked herself through the surgery hundreds of time, when she can’t sleep at night it’s the first thing that her mind drifts to, there isn’t any set routine for a laparoscopy, it’s based on how much tissue has built up and where it’s built up. They won’t know until they open you up, from the limited amount of information Alexia has been able to extract about your past surgeries she’s figured out that for you in the past, it normally builds up in your pelvis, fallopian tubes, the ligaments around your uterus, your ovaries and the last time you had a laparoscopy they even found small clusters on your bladder, vertebraes and bowel. Hopefully, they’ll just find it around your uterus, but if there are signs that it might have attached to other organs then the surgery can become far more invasive and longer. Your doctor has said that in the best case scenario it could take an hour or two, worst case scenario it might take upwards of 5 hours.
At least two hours of Alexia falling deeper and deeper into her fear that something horribly bad is going to happen.
You’d encouraged her to touch base with her therapist in preparation, a proposal that Alexia had laughed at. She knew that you’d talked to your psychologist a day beforehand in preparation, and told her that if possible you’d like to follow up a few days after your surgery. Alexia had no reason to reach out though, she wasn’t the one who was having her reproductive organs sliced up, it was her job to make sure that you were able to achieve optimal recovery. Your health was Alexia’s biggest concern, she needed you to recover, she needed you back on the pitch with her, she needed you back to normal as soon as possible, so that all of her stress and anxiety about everything that was happening could dissipate.
“Alexia, you’re squeezing.”
Alexia looked down at her hand, it was resting on the outside of you thigh, her hand all fisted up in your skin and the paper thin hospital bedding.
“Sorry bebita.”
Alexia retracted her hand as if had been burnt, she felt horribly guilty about the selfishness of her feelings, it wasn’t right for her to be so twisted up in her own troubles. She wasn’t the one suffering.
She didn’t know how to explain it, she felt like a child with her hand between couch cushions, reaching for something and only grazing for the edge of it. She only grazed the edge of understanding what you were going through, it felt like she was eternally reaching for more, trying to feel it, and yet she didn’t have hands big enough to grab what she wanted. She didn’t have the tools to fix this, she didn’t have the body to understand what you were going through, she was left searching without any chance of retrieving anything.
“Alexia.”
Alexia looked up at you, there was something in your body language, like you were uncomfortable, or on the border of being uncomfortable.
“I’m sorry bebe, is everything alright? Are you comfortable? I’m sure they’ll be in soon to come and get you, I know you must want to get this over with.”
You roll your eyes, and Alexia is aware that it’s your way of saying no, but for some reason it cuts deeper for her, she’s so on edge, she’s been on edge ever since you got your surgery date and it’s all starting to reach a certain point that is pushing Alexia in ways she doesn’t understand.
“I want my girlfriend to relax. I’m going to be fine Alexia, nothing is going to happen.”
Alexia knows that factually you’re correct, but in Alexia’s imaginative brain she can’t wrap her head around the truth.
“I know bebita, I just want to make sure that you are feeling good, si? You’re comfortable? It’s important for your recovery that you feel as good as possible.”
Your roll your eyes again.
“Alexia, everything is perfect, it’s all fine. I’ve recovered from this same surgery two times, I’m prepared. You’ve made sure that when I get home I’ll be comfortable, I’ve walked you through what the next week might looked like. I’m prepared and so are you, everything is going to be fine.”
You’re eerily calm and Alexia can’t tell whether or not it’s a front to hide your secret insecurities or if you seriously are content with what you are going through.
She supposes that when you’ve been living with something so debilitating for such a long time that at some stage you get used to constantly being disappointed with your body and constantly knowing that it doesn’t perform the same way that other people’s do.
“I know bebita.”
Alexia doesn’t even sound like she believes herself.
“Do you? Because it doesn’t seem that way.”
Alexia doesn’t want you worrying about her, she doesn’t want you worrying about anything, she wants you to feel calm.
“Bebe, I’m fine. I know our plan, I was the one who made it.”
It’s a weel made plan. Alexia had spent weeks getting everything in line for this moment. You were both going to take at least a week away from football, two weeks if need be. Alexia would return after the rest period, to training until she felt like she was in a place to be back on the pitch. You’d come in with her on the good days, meet with the team doctors and work through whatever exercise that was allowed whilst also just reintegrating into the team. If it all went to plan you’d make your way back into team training at the six week mark, and if it all went to plan you’d be back playing around the eight week mark. For the first week, you would be on mostly bed rest. Alexia had already meal prepped your comfort foods and easy foods to digest for the first week, and she had no doubt that her teammates would drop some more food off. You would rest, allow yourself some grace to recover from what you were going through. Alexia would make sure of it. She didn’t know what you’re previous recoveries from your surgeries looked like, you hardly shared any details about your past with endometriosis but what she could gauge it musn’t of been a positive experience. She was determined to change that though.
“You’ve got nothing to be scared about Ale, I’m going to be fine. I can’t guarantee how happy I’m going to be once this is over but I’ll be okay. I’ve been here, I’ve done this before, it’s the same as every other time.”
Alexia doesn’t think she’s seen anybody go into surgery so casually, as if this is truly just another surgery for you, as if there is no magnitude behind what is about to occur.
“I know bebita, I know.”
Alexia reaches for your hand, giving it a squeeze before bringing it up to her mouth and pressing a soft kiss to it. Her lips linger for a second, it feels good to know that you are here with her.
“Ms Putellas, it’s time to take her.”
Alexia looked up to the doorway, where one of your nurses was waiting.
She didn’t want to say goodbye to you, she didn’t want the waiting to start.
Alexia stands up, moving her lips from your hand and pressing them to your hairline, breathing in everything about you. The smell of your shampoo that she’d massaged into your hair this morning, the smell of your body soap, the warmth of your skin. It was all you, the you that Alexia knew, the you that Alexia loved so much. She had so many hidden insecurities, but her biggest one was that deep down this surgery was going to somehow change you, and when you came back different to how you currently would, you wouldn’t want her anymore.
There was no explanation, but it didn’t change what Alexia had come to believe.
“Alexia, I love you, I am going to be fine, I will see you in a couple of hours.”
Alexia didn’t know it, but you had your own insecurities. You’d never had anybody, friends, family, partners that had stuck by your side through all of your issues. Who was interested in a person who was completely broken from the inside? Why would Alexia want to stick around for all the troubles. It wasn’t easy looking after a person with a chronic disease, even with her limited amount of time dating you she’d hardly experienced the real struggle. Sure, she’d seen you on your period, but that didn’t compare to a lifetime of struggles.
Alexia had advocated for you, with the team, with the physios, with the coaches, with your doctors. But was she prepared to do that for the rest of her life? Was she prepared to sacrifice her own health at certain times to prioritise yours? It was the ultimate sacrifice, and even though Alexia was genuinely the best person that you’d ever met, yet you wouldn’t blame her for being unable to compensate for you.
“I know bebe, I’ll be waiting for you. I love you so much, it’s all going to be fine.”
Alexia pressed her lips to your forehead one final time, you didn’t push her when she took her time. Alexia’s lips shook against your skin, the quiver silent as she kept any emotion she had at bay. She needed to be strong, she needed to stay strong for you.
“Ms Putellas, one of the nurses will take you to the waiting room, a doctor will come get you when she’s done.”
Alexia finally pulled her lips away, it truly did feel like she was parting with you completely, like she was being pulled away from you completely.
The nurse reached for Alexia’s hand and she shook it free.
You frowned at her, like she was making this so much harder, it made her feel bad, but she needed this, she needed a moment.
“I love you.”
You let up on your frown.
“I love you too.”
Alexia nodded, if there were tears in the corner of her eyes then she had no idea, her focus was on you, only ever you.
Only when she finally felt comforted, did she let the nurse lead her out.
The nurses took you down one end of the hallway, and she dragged herself in the opposite direction.
The waiting room was the same as any other, random stock pictures of scenery from somewhere around the world, chairs that were more uncomfortable then any other surface, bright white lights that were bound to remind Alexia exactly where she was.
It was all uniform, besides her mother sitting in the corner.
Eli looked the same as she always did, her lips sitting in a content smile. There were big lines all over her face from her smiling all the time, it was the reason why everybody gravitated towards her.
“Mami.”
Alexia had known there was something slightly suspicious happening when her mother had been dodging her calls for the last few days. She’d assumed in typical fashion that whilst Alexia was looking after you, Eli would go into overdrive, cooking all the meals, helping Alexia. She hadn’t quite expected this.
Alexia had kept the details about your surgery vague with her family, she didn’t feel like it was her information to give away and also she wanted to keep some privacy. She knew it was going to be a tough period for you and also for herself.
“No, before you say anything, I am staying.”
Alexia smirks, pushing back the words that had already been on the tip of her tongue.
“You don’t need to be here, I could be waiting for while. I can call you when she’s done and you can come to see her.”
Alexia walks towards her mother, and she knows by the way that Eli is looking at her that she’s not going to be able to convince her to leave. Alexia’s stubbornness came straight from her mother.
“I will do no such thing. I shall wait here with you, I’m here to support my daughter.”
Eli nudges Alexia into the seat beside her.
“I’m not the one having surgery Mami, I will be fine waiting on my own.”
Eli slaps Alexia’s arm.
“Estupida. Tan estupida. When I was in hospital with your faher do you know how alone i felt? No, you will not be alone. Maria has gone to go and get coffee, we will all wait here together until she’s done. Comprendida?”
The tears that were in Alexia’s eyes a couple of minutes ago are back, the sick feeling of anxiety that’s been in her stomach all week, especially today suddenly feels less all consuming.
“You don’t have to do this alone, there are people here for both of you. There is support for you.”
Alexia nods, and for the first time in a while she actually feels like she doesn’t have to handle it all, like maybe she can off load some of the responsibilities that are making her chest ache with every breath she takes.
“Thankyou.”
As the doctors had told Alexia, the first week is by far the worst.
Your surgery ends up going for a lot longer then expected. As the doctors had expected, it hadn’t looked good, and it had spread to some of your organs. The positives were though that there hadn’t been any complications and they’d managed to remove all the affected tissue.
It had ended up being a day procedure and the doctors had cleared Alexia to take you home once you’d woken up and were eating and talking.
It had only gone downhill from there.
You were in a lot of pain, and pretty much delirious on the feeling of it and anaesthesia.
Getting you out of her car and into the house had been near impossible, carrying you was too uncomfortable for your incision, and walking was also impossible.
Eventually, with a mixture of limping, Alexia bearing all the weight for you and dragging you as carefully as she could she got you into your bed.
Alexia had set the house up so that when you were home everything would be easy, she hadn’t organised though for you to be completely immobile.
She got you set up underneath the blankets, allowing you to rest whilst she worked on making a light meal for dinner.
When Alexia came back, you were running a fever.
It was a low grade fever, something the doctors had told her to expect but it made her feel uncomfortable.
A fever was the first sign of an infection, and you weren’t awake enough to know what was going on. What if something bad was happening and she didn’t know? What happened if she didn’t catch it early enough? What if she was the reason that something bad happened?
Before Alexia could find the warning signs of slipping into her own brain she was falling deep into a fog of anxiety.
Alexia was the one who was responsible for making sure that you were okay. She thought that she’d worked out every different variable and issue that could arise, she’d promised you that she’d be there for you after your surgery. If she couldn’t help you with this though then was she really capable of dealing with all of it?
You stirred slightly as Alexia removed the thermometer from your lips, your eyes hazily opening up, looking at her with confusion.
“Hugs.”
It’s almost laughable the way that you whine at Alexia, the doctors had said that you would be pretty out of it for the first few days. You’d told Alexia that in the past, the first two or three days after your surgeries tended to consist of mostly sleep and adjusting to the pain of your healing incisions. After they had healed it was all upwards from there.
“How are you feeling? Do you want some more pain relief? Do you need some water? Food? You’re not feeling nauseous or cold?”
You shook your head, your lips morphing into a soft smile as you took in Alexia, all flustered and clearly worked up.
“Want hugs.”
Alexia was guarded, she always was when she was focused on protecting people. You didn’t want guarded Alexia. You wanted relaxed, homey, calm Alexia. The Alexia who would sit on the couch with you after a rough game and eat icecream, the Alexia who would silence her alarm on a morning workout day to stay in bed with you, the Alexia who just wanted to spend all of her time with you. You just wanted the Alexia that would put whatever you wanted above all else.
Alexia looks torn, really torn, like she knows that she should probably be caregiving in some way. She promised you though that she’d let you guide her through your process, that she wouldn’t try to control how you wanted to recover.
“Okay bebita, if that’s what you want.”
Alexia slips to the wardrobe quickly, pulling off her sweats in favor for changing them in for some softer pyjamas. She doesn’t know how long she’s going to be in bed with you but she assumes it’s going to be long enough that she’ll want to be comfortable.
When she makes her way back into the bedroom you’re waiting patiently, like you know that if you’d fallen asleep whilst Alexia was gone she probably would have gone off to tackle some non-existent chore to make herself feel better. Alexia rolls her eyes at how well you know her more unobvious traits and tells.
Alexia, without trying to shift the mattress at all, pushes herself under the covers of your bed. She’d made sure to put down fresh sheets, so it’s comfier and snugglier then it normally is. When you realise that Alexia is going to make no attempt to shift within cuddling distance you pull your arms out from underneath the blankets. Everything is sore and accompanied with a tinge of pain that makes your head hurt. It sucks. You want your teddy bear of a girlfriend to hug it away.
Even though it hurts, you reach for the hem of her sleep shirt, tugging it towards you. There is no power to your movements but there is a general message. Alexia shuffles closer to you, but still far away enough that you definitely can’t use any part of her body as a pillow. You don’t know why it hits you then, you know you’ve got a lot of built up emotions, and that hormonally there is so much going on in your body and the pain medication isn’t making it any better. You don’t know why, but all of a sudden the tears start to fall and you can’t contain the sniffle that you make.
“Bebita, what’s wrong?”
Alexia shoots up, ready to take action in whatever way is needed.
“Do you even love me? What person wouldn’t want to give their girlfriend a hug? I know I just had surgery and I’m digusting and haven’t showered and smell like hospital but can you not just give me a hug, is is that big of a ask?”
Alexia almost immediately recoils, the feeling that sets into her stomach is one of horror. Of course she wants to hug you. Of course she wants to love you, she’s just scared to hurt you.
“Bebita, no. It’s not about any of that, I just don’t want to hurt you bebe. I love you so much bebe, none of the surgery stuff bothers me, not in the slightest. I just don’t want to hurt you.”
At this stage you’re sobbing hysterically and Alexia feels so horrible.
“I know you didn’t want me to get the surgery, I know it was stupid but can you just hug me?”
Alexia genuinely does a double take when you say that, it’s a insecurity that you haven’t expressed to her at all and she doesn’t quite know what to make of it.
“Bebe, of course I wanted you to have the surgery. I can’t tell you whether or not it was a good decision for you, but for me it was a no brainer. Bebe, you were in so much pain, you were struggling so much and for me it was so hard to see you go through that. I don’t know why you’d think that I wouldn’t want you to get the surgery but I can tell you that any decision that makes you feel happier and healthier is always going to be the best decision. If this surgery makes you feel healthier and happier then why would it be stupid?”
You don’t really know what you are thinking or why you’re suddenly thinking all of these things but you just feel so wrong all of a sudden.
“You promise?”
Alexia smiles at you, finally leaning over into your personal space. She bring her hand up to yours and interlocks your pinky with hers.
“I pinky promise, now how about we work out that hugh that you want. I don’t want to hurt you, so what’s going to work best?”
Alexia reaches up to thumb away the tears that are still leaking down your face, you can’t control the random emotions that are flurrying through your body like a tornado.
“The doctor said I should sleep propped up, so I could lie with my back pressed to your front?”
Alexia figures that probably is the best way to approach this whole situation.
It’s not easy getting you to sit up for long enough for Alexia to slip in behind you, but you both reap the reward when you get to relax against each other.
Even though you’re wrapped in Alexia’s hoodie she swears that she can feel your bones relax against hers and even though she knows that you might be in pain she selfishly couldn’t care less. For the first time since she’d walked through the doors of the hospital this morning she finally feels like she can breathe, like she has you, like her inherent need to protect you from everything has finally been fulfilled.
Alexia worries about you more then anything else, when you get injured, when you’re sick, when you’re away from Alexia too long she worries. Her need to protect and care for you has only developed more over the course of your relationship, and when she found out about your endometriosis it doubled. She’s never worried much about anything besides football, and even when she worried about football it was never the same kind of worry, it’s more internalised stress about achieving what she’s set out. For you, everything is irrational, none of it makes any functional sense, it’s ridiculous. And yet she kind of loves it, because she knows that it means it’s all real, that her love for you is so real and true that she worries irrationally because the thought that anything bad could ever happen to you she knows that it would break her in ways she’s never been broken before. She saw it in here mami when here papi died, the way that she stopped being able to function in the absence of him, and god forbid anything similar happened to you Alexia knows she’d be the same.
There’s this thing about recovery, about recovery from a non-injury that makes your skin crawl. You’re not a good person to be around when you’re struggling, it’s the half the reason you’d kept your reproductive issues from Alexia, because it all made you grumpy and tired and borderline and fucking bitch. There was a pattern of shutdowns, emotional, physical, mental, every single time you had a slip or bad period. A surgery though, it was a whole different story.
By week two you felt pretty much back to normal, but as the doctors kept reminding you, you weren’t. That was the problem with internal surgery, eventually the outside of your body returned to feeling normal, but the healing process on the inside took ten times longer. Your scars were healed, even the swelling had started to go down, the pain was minimal but you were still healing.
By week 4 you knew that you were toeing the line of being a bitch and being intolerable. You stomped your way around the halls of the training facilities, annoyed with your lack of progress and the fact that whilst all of your teammates were out on the pitch, you were stuck indoors under the air conditioning doing the exact same pelvic floor exercises that you’d been doing for two weeks. You were now allowed to run on the treadmill, instead of the slow walk that you’d been doing for the past few weeks. It all felt like nothing though, like you were being pulled through hell with no real picture of when you would be allowed to return.
You’d known about all of this before your surgery, known that it was going to be a slow process. What you hadn’t realised though was just how lonely that process was going to be, nobody understood what you were going through. At least with injury there was common ground, everyone on the team had been through some kind of back setting injury that had them in the gym more then on the pitch. They’d never been sidelined for something inside of them though, something that was going to keep coming back and keep hurting them over and over again.
Week 5 was when you broke. There wasn’t any exact reason, maybe it was the way that Mapi continuously kept trying to get you to try some stupid gym challenge when she knew you weren’t allowed to, maybe it was the way that Alexia kept coming home everyday complaining about how the defenders couldn’t get their act together, or maybe it was just the overwhelming pressure that was starting to get to you.
All you know is that after a particularly hard day and Alexia groaning about the sprints they’d run after every drill on the way to the car, it had been the final straw for you.
The minute your seatbelt was fastened there were tears pouring down your face.
When Alexia reached out you smacked her hands away.
“Bebe, talk to me, are you in pain? What hurts?”
If you weren’t busy crying you’d probably slap Alexia for the stupidity she was exhibiting.
“No, no I’m not fucking in pain. I’m sick of you constantly fucking tell me about all of your problems and acting like I wouldn’t die to be doing what you’re fucking doing. I’m sick of everybody thinking that I can do everything that they are doing just because I look fine. I know I look fine, I know I seem fine, hell at this stage I’m pretty sure I am fine. I’m sick of feeling like I’m back to normal and being told that I’m fucking not, that I have to wait another week, that I can’t do things that are so normal to my life.”
Once again when Alexia tries to give you a hug you deny her, wiggling out of her hold as soon as she tries.
“Bebe, no recovery is linear or easy. Just because yours looks different to most peoples doesn’t make it any less valid. You need to be more gracious to yourself, you are going to get back from this. Just because it isn’t a normal process or what a recovery normally looks like for you doesn’t mean that you are any less entitled to your process.”
You know that like everybody else, Alexia doesn’t understand. Sure, she’s injured herself, she’s been through hell and back for her recovery. She doesn’t understand though what you feel, the immense struggle that you are going through.
“No Ale, you don’t fucking understand. There was no point in your recovery where you weren’t able to eat your favourite foods because your body didn’t react well, there was no limiting you from sex, there was no limits on what the unaffected parts of your body could and couldn’t do. There was no limits on how much time you could spend in certain positions or limits on whether or not you could fucking vaccum or carry something heavier then 5 kilos. You don’t understand, I have nothing to look forward to, i’m going fucking crazy watching everybody do things daily with ease that I can’t fucking do. My fucking body hates me, it betrays me everyday. And I can’t change that, I can’t do anything about that.”
Alexia forces herself to take a deep breath, to truly consider what you’re saying. She doesn’t want to invalidate you, not even remotely, she knows how easy it is for people like you to be invalidated because of your disease. She has no idea what you are going through, she can’t even begin to understand, and yet she’s been trying to. She’s been trying to empathise with what you’ve been feeling, she’s tried her hardest to force her feet into your shoes but the reality is that she will never be able to. Nothing is going to make her understand the magnitude of what you are going through, and she thinks that she needs to stop pretending that she does.
“You’re right bebita, you’re right. I’m so sorry, just tell me what you need, whatever you need and I will help you. I will try my hardest to do whatever it is you need. Just take some deep breaths with me, vale? It’s okay, the world isn’t going to end right here and right now, it might feel like it but I promise it won’t. These are fixable problems, we can fix this, maybe not right now which sucks but eventually this will all be fixable.”
This time when Alexia moves to hug you, you don’t flinch away. For the first fucking time in five weeks you be slightly heard, less alone and isolated.
You sob into Alexia’s neck, not caring that it’s making her neck wet and also your face. It feels good to let it all out, like all of the weight, all the words from your psychologist, from the doctors, from the physios, from your teammates, from the coaches, from Ale are slowly becoming background noise. All you want is for it to be background noise, for it to all disappear.
“We’re going to be okay bebe, you’re going to be okay, we’re going to work this out.”
Alexia doesn’t really believe her own words, she doesn’t know if everything is going to be okay, not remotely. She doesn’t know how to help you, after you became self-reliant after week 2 her job as your caregiver was pretty much nonexistent, but that didn’t mean that you didn’t still need help. It was all confusing, a back and forwards dance that didn’t really make any sense really. Whether Alexia had realised it or not though, she’d assumed that you were happy to do it alone, inadvertently. She was giving you your space to recover how you wished, even though it was clear that wasn’t necessarily how you wanted to go about your rehab.
When your cries softened Alexia let go a little bit, so she could look at you.
“How can I help bebita? Tell me what I need to do and I will do everything I can.”
Your eyes are all glassy and full of tears, it makes the hole of guilt in Alexia’s heart even bigger.
“I just want to play.”
Alexia furrows her face, because of all the things she can do, that isn’t one. She can’t clear you to play and she won’t condone you rushing your rehab to get back. She wants you to play, do desperately, but only when you’re at full health.
“I can help you with your rehab, bebita you can be cleared from the six week mark onwards, if you work really hard this week there is a chance you could be back in training next week. But you can’t rush yourself, I know you want to be back on the pitch, I want you back on the pitch. But you need to be ready, and right now you aren’t.”
Alexia thinks it might be a little bit too brutal, but the way you nod at her with real determination tells her that she’s said the right things, she’s supporting what you need whilst also being honest.
“I can’t get you back on the pitch but how about icecream, huh?”
You look like you might start crying again but after a few seconds you roll your eyes in the same way that you would to Alexia on any other day.
“It’s a wednesday, your cheat day is Mondays.”
Alexia shrugs, because cheat days aside, she’d break any and every rule just to see a smile on your face.
“Well last time I checked everyday is a icecream day, especially on wednesdays.”
Alexia’s goofy smile never leaves her face. It’s clear on your face though that you’re still worrying.
“Bebita, you can’t do anything about this. You just keep doing what you’re doing, even though it sucks and we wait until your six week appointment and hopefully you get good news. The doctors said if everything went how it should then next week should be when things start to pick up, you just have to trust the process bebe and it’ll all work out. Until it all works out, we have icecream, no?”
Your barriers break and for the first time since you’d sat down in the car you smiled at Alexia.
“Yes, we do have ice cream.”
Alexia had been forbidden from coming to your six week appointment, you wanted to do it alone, wanted to face whatever news you were given on your own and she respected that.
It didn’t make it any easier.
In the space of around an hour she had reorganised her wardrobe and then reorganised it again because she couldn’t handle not having certain things in certain places. She then moved on to folding all of the laundry, making sure her kit bag had everything that was necessary for the away game coming up and then out of pure desperation she moved on to cleaning all of the dishes.
She’d seen the way that the last week in particular had drained you, how everyday had been even more of a struggle. For the first few weeks, there had been moments of intimacy, moments of love and softness. All Alexia wanted was that back, she wanted to be able to love you and have you reciprocate it without withdrawing. She just wanted some part of you back.
When she heard your keys turning in the door, she tried her hardest to look normal, perched on the kitchen bench like a kid waiting for their parents to come home from work.
She listens to the sound of the door clicking open and shut, then the sound of shoes scuffing off, a bag being dropped and then your footsteps. Alexia’s heart beats faster and faster as she listens to you slowly get closer to her.
She’s rewarded when she finally sees you, and you’re not crying and don’t look like your about to punch something so she thinks there must be some kind of good news on the horizon.
When you spot Alexia clearly waiting for you, you almost laugh, she looks like she’s trying so hard to not ask you what she clearly wants to. She looks almost antsy.
“Hi.”
You keep walking, closer and closer, until your standing in between her legs and looking up at her from her spot sat on the counter.
Before she asks a question, you lean in, you need it. You need to feel Alexia in a way that’s been forbidden for far to long.
The kiss is breath taking, not to rough, not to over bearing. It’s just enough of everything that you feel like you’ve been missing out on. It makes everything better.
Alexia lets you take control, it feels good to not be out of control with everything that’s been happening.
Eventually though, when your hands move from her neck to the hem of her shirt, she pulls away.
“Bebe, I love you, but if you haven’t been cleared we should stop here.”
You roll your eyes and fish your hand into you back pocket, pulling out a folded up slip of paper.
“I think you’ll find that you have a girlfriend who has been cleared to return to all activities except games until I’ve gotten sufficient training hours, I’ve got the green light amor.”
Alexia pulls the paper from your hand, her eyes scanning every single word of the clearance form your doctor, making sure that you’re telling the truth. When she gets to the bottom, her face lights up and before you can say anything she’s standing up and lifting you in her arms.
“Alexia, put me down.”
You giggle as she balances you on her hip as if you are much smaller then you truly are, already walking towards your bedroom.
“I think it’s time to show my beautiful, patient, hardworking girlfriend just how much i love her, unless you disagree?”
You have no objections.
Alexia drops you down on the bedding as softly as possible, a big cheshire grin plastered across her face, like a huntress whose found her prey.
She joins you on the mattress, crawling up your body and kissing different places as she goes.
When she get’s to your head she presses her lips to yours, there isn’t any inhibitions or barriers, she kisses you like a person whose been so deprived of something that they’re craving it, like she’s greedy for you. Alexia moves her lips everywhere, your neck, your collarbones, as low as she can get until your tank top stops her and when she wants more, she gently tugs the top off.
Alexia worships your skin like it’s holy, she kisses and sucks and bites until your sweating and so worked up that you can’t think. When it starts to get to you, she unclips your bra, and that’s a whole other level of worshipping every inch of breast and then your nipples.
It’s glorious, 6 weeks of celibacy had been hell. Technically this would have been okay, but neither of you trusted yourselves to get to this stage and not go further.
So being here, being at this level of pure ecstasy and pleasure was everything you’d wanted and it made all of the tough moments feel so much smaller.
After she was sufficiently happy with her worship of your upperbody, she began her campaign downwards, spending time on your ribcage and stomach before spending extra time showering your incision with soft kisses.
It was something she’d done regularly since the bandages had come off, everytime you stared in the mirror for two long, or complained about how ugly it looked she would spend at least half an hour reminding you just how perfect your body was with certain scars. Those scars were proof that you were healthy and taking care of yourself and for Alexia that mattered far more.
From your incision it was down to the band of your leggings, which after a quick glance up to check you were still happy, she slowly peeled off.
“Alexia, enough teasing.”
Alexia pressed open mouthed kisses all along the outline of your panties.
“I’ve been away from your body for so long bebita, I’m just reintroducing myself.”
You rolled your eyes at the blonde, and were about to say something bratty but were stopped when she moved her open mouth a bit further down, just above your mound.
Instead of a bratty retort Alexia’s ears were met with the glorious sounds of little moans slipping from your lips as she teased you.
When you started bucking up into her, Alexia made the executive decision to move things ahead a bit quicker then she intended, after all she had plenty of time to do this, over and over again.
“You’ll tell me if anything hurts, si? If anything feels wrong say something, we’ll just stick to my mouth for now and work our way back up.”
You nodded at her, even when you felt good sex could be painful, it was a weird back and forth of you feeling good and then all of a sudden feeling bad. All you craved right now though was sweet release.
“Want to taste you bebita, god I missed tasting you.”
You swear your legs go numb and you almost orgasm just from her words.
“It might be quick, it’s been a long time Ale and I’m sensitive.”
Alexia smiled at you, kind and genuine like she had all of the time in the world.
“Just feel it however you need to, if it’s fast then it’s fast, whatever you feel just take it.”
You nod at your girlfriend before relaxing against the pillows on your bed, allowing Alexia to pull one of your knees above her shoulder.
The feeling of her mouth on you is truly the best feeling in the world, you don’t care how loud you are because the fulfillment you are getting has been so hard worked for that you can’t help but just enjoy it.
Alexia licks and sucks in all of your most sensitive places, places that have been untouched for so long that you’d almost forgotten just how good it could feel. When she pulls your clit into her mouth you see stars and you know that you aren’t going to last long at all, you gather Alexia can tell from the way that your octave heightens.
“Ale-Ale I’m going to cum.”
You can feel Alexia smirk against you.
Even though she knows that she’s able to satisfy you, there is nothing better then actually doing it.
“Go ahead bebe, I’m right here, you’ve been so good and patient. You deserve it, if you feel it just let go.”
You nod your head, biting down on your lip as you feel yourself getting closer to the edge, there is no hesitation from your edge, when she grazes her teeth against your clit you are well and truly gone.
Normally, you’d last more then one round, or you’d reciprocate at the least but before you can even try Alexia has you bundled up in her arms, pressing kisses all over your face.
You squeal and flinch but she doesn’t stop, not until your face is all red and your both relaxed against your pillows.
“You know how proud I am of you? You’ve been so good this whole process bebita, so perfect, so hardworking, so determined. You’ve pushed through it all, and you’re going to keep doing ti because your the toughest person I know, truly. I love you so much amor.”
You know that for the rest of your life, Alexia will take care of you, like nobody else. Underneath the surface she cares for you more than anybody else ever had and ever will.
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iindiafterdark · 3 days ago
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you oughta know
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summary: you were failing your statistics class, so your professor recommended you to abby for tutoring. after the first session, you left with not one, but two lessons.
pairing: nerd!abby x mean f!reader
warnings/tags: 18+ mdni, college!au, no use of y/n, dom!abby, sub!reader, swearing, fingering(r!recieving), dirty talk, degradation kink, exhibitionism kink, dumbification, breath play
wc: 2.1k (slightly proofread)
an: shoutout to my ghost writer @only4theweeknd for fueling me with this idea, it's very freaky but i hope you guys enjoy
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you huffed dramatically, sauntering into the poor, dimly lit library of your college. long story short, you had a c in statistics, and you were in desperate need to raise your grade. but, to be quite honest, you couldn’t care less about the actual class, as you found yourself on your phone during most lectures. but you cared about what your parents thought, so you had to do something.
your professor recommended you to abby, one of the smartest statistics tutors on campus. you’ve seen her around campus a few times, mostly quiet and wedged in a corner, keeping to herself. you rolled your eyes at the thought, irritated that you’d probably spend this hour and a half listening to her nerd out over some formulas and graphs that made no sense.
you scan the mostly empty library as you walked in, looking for that specific blonde-haired braid abby always sported. you walk the perimeter of the library, finally stumbling onto abby sitting at a table lodged in the corner with books and papers scattered across it.
she looked extremely focused, her brows furrowed deeply as her eyes rapidly explored the pages in front of her. you crossed your arms before clearing your throat loudly, getting her attention. her green eyes snapped towards you over her thinly framed glasses, riddled with confusion. her puzzled look made your growing irritation sprout even more. 
she studied your face in a minute’s silence before her eyes flashed in recognition. her hands shot out to the table, clearing up most of the documents before meeting your eyes again. “oh, it’s you. the one i’m supposed to be tutoring for professor jackson’s class. i’m abby, nice to meet you,” she stated, her voice void of tone. “yeah yeah, likewise. let’s get this over with, please,” you grumble, pulling the neighboring chair back and plopping into it.
she didn’t react to your brewing attitude as she calmly organized the table a bit more. you slid your laptop onto the table, waiting impatiently for her to get everything settled. your eyes settled closely on her appearance. her neatly ironed white button-up and black slacks fit her perfectly. 
her braid slung over her shoulder, stray hairs framing her stoic face. you zeroed in on her fingers, noticing how thick they were. and the way they slid over the papers, nimbly. your mind couldn’t help but wonder what they would look like-
her hushed voice mumbled your name, snapping you out of your shameless staring. “let’s get this started, yeah?” she suggests, giving you a shy look.  heat rushed throughout your body, the feeling of embarrassment creeping up on you. you give her a swift nod, opening your laptop to avoid having to look at her.
it was about twenty minutes into the session, your irritation had subsided slightly. abby was talking to you about the recent unit, explaining something about probability distribution. your mind was partially elsewhere, thinking about whatever party you were missing tonight to be here. 
“so..the discrete probability distribution has random variables that have specific separate values?” you question, tilting your head to her. you were still confused, no doubt, but you were paying some attention. she glanced your way, a small smile on her face as she nodded. “yes, that’s right.”
your brows furrowed even more, although you understood a portion of what you were being taught. “but, if random variables are random, then how can they have specific values?” you added, giving her a bewildered look. abby brought her hand to her face, sighing into it before leaning back in her chair.
she returned your gaze with a heated one, irritation swirling deep within her irises. “maybe, if you actually paid any attention to what i was saying, you’d know how that works,” she quipped, crossing her arms.
you scoffed at her remark, although you were distracted you were paying some attention to what she was trying to teach you. “well guess what, i didn’t even want to fucking be here. i could be at a party right now instead of being in this worn-down fucking library with you.” you spit, the words sliding off your tongue like cursed venom. 
abby kept her composure, her eyebrows raising in what looked like astonishment. you held her gaze, feeling the heavy, sparking tension rising the more you stared at her. she adjusted her dress shirt, rolling the sleeves higher on her muscular arms. if you weren’t so incredibly annoyed, you’d be gawking at the sight of them. she leaned closer to you, your breath hitching as you tried to hold your ground. 
“maybe if you weren’t such a fucking hard ass, we’d actually get something done, hm?” her breath ghosts your cheek, making your thighs clench in tandem. her quiet, shy demeanor was long gone, leaving the subtle assertiveness and looming dominance in its wake. 
she didn’t wait for any of your snappy remarks, wrapping her hands around your wrists to haul you up from your chair. you gasped as she manhandled you, placing you in front of her.
she places her wide hand over your back, pushing you down slowly onto the table in front of you. your heart was beating out your chest, your breathing uncoordinated as you tried to catch your bearings. but god, the ache in between your thighs was unbearable. your thighs trembled with both fear and insatiable need. 
once your chest grazed the cool oak table and your cheek pressed against her papers, abby’s hands began to wander. they slid towards your ass, palming it before giving it a harsh slap that rang in the small corner of the library. you jumped, a small whimper tumbling from your lips. a low chuckle reverberated from her chest, riddled with sick amusement at your oozing submission. 
“you didn’t like that lesson, right? it’s okay, i’ll teach you another lesson instead.”
your cunt throbbed eagerly at her words, the want that settled deep in your bones coming to a rearing head. you cursed yourself for wearing a skirt, especially now that you could feel your slick soaking through your panties all for abby to see. 
she continued to explore your pliant body, grazing and squeezing your plush thighs in her hands. she did it in silence, letting you sit in the uncomfortableness of being so vulnerable, so open to her. all the thoughts in your brain faded, your mind becoming blurred, fuzzy as she teased you. you hadn’t noticed that you were pushing your hips back, desperate to get something, anything from her.
she let out a breathy laugh, giving your ass another hit before rubbing the stinging pain away. “oh, don’t be so impatient. that’s what got us here in the first place, isn’t that right pretty girl?” she simpers, swatting her hand down. you nod, letting out a high-pitched whine as she spanked you once again. 
“don’t just nod at me, use your words.”
“..y-yes, abby. you’re right,” you gasp, reveling in the way her hand trails to your damp, clothed cunt. she hums in satisfaction, her fingers rubbing small circles over your clit. 
your back arches more in response, leaving yourself completely bare to her antics. “that’s right, baby. impatience got you bent over this table like a slut with this pretty pussy leaking in front of my face,” she teased, hooking her finger around your panties, pushing them to the side to reveal your leaking slit.
“oh fuckkk, you like this huh? like getting spread out in the middle of this library f’me?” she taunts, using her fingers to spread your cunt open, your arousal dribbling down her hand. you attempt to grip the table, only to be met with the plethora of papers on it. 
you didn’t want to admit it, but the idea of someone catching you, seeing you like this for her made your cunt clench even more. “ngh..yes- god yes i love it, please,” you mewl, moving your hips to push down on her fingers. biting pain shot through you as she struck your ass, the movement of it making her groan. 
“good sluts use their words, baby. tell me what you want,” she urges, ghosting her fingers over your bare cunt once again. her fingers toyed with your folds, the soft squelching sounds making you leak even more as you listened. your ego and conceit slipped through your fingers, only leaving you to beg like a deprived puppy. “p-please abby, i want..want you to finger me,” you beg, your voice hoarse, quiet only for her to hear. 
“how ‘bout a pretty please, hm? can you do that for me?” a condescending lilt twisted within her tone. you shot her a fiery glance, only for her to slowly pull away from your cunt, making your eyes go wide.
“no! fuck..okay. i want you to fuck me on your fingers..pretty please?” you mumble, feeling the last remnants of your self-respect crash and burn. abby only laughs before finally slipping her two fingers into your sopping cunt. the loud squelch is deafening, your wetness making a ring around her fingers as she plugs your needy hole. 
you keen at the contact, feeling her burly fingers fill you. your cunt swallows her fingers in earnest, clinging to them tightly as she pumps them. “shiiit, you take me so well, baby. can barely move ’em without you clamping down on me,” she groans, her voice making your hole flutter around her even more.
she finds your sweet spot almost instantly, poking and prodding at it with every push of her fingers. your futile attempts at holding onto something to ground you have dissipated, only leaving you there to drool against the worksheets you were supposed to be learning. 
she notices, of course, looking to see the clear wet puddle of spit soaking through the paper. her fingers thrust deep into you, firm and unrelenting as she laughs at the pitiful sight in front of her. “you’re drooling on my worksheets, you know. you don’t care though, do you? just a dumb slut that wants to get fucked on my fingers,” she teases, pressing the pads of her fingers into your g-spot hard.
“f-fuuuck abby please! mmngh- please oh my god,” you moan, your words slurred and incoherent. you don’t even know what you’re begging for at this point, just babbling as your brain turns into mush. 
your pretty moans fuel her even more, thrusting her fingers into you at a wild pace. “c’mon pretty girl, say it. tell me what you are,” she demands, rising out of her chair to wrap her other hand snugly around your throat. she’s not choking you no, just enough to make you understand the hold she now has over you.
your desperate pleas and moans have devolved into needy pants, your brows furrowed as you feel a tight build in your stomach. you begin to fuck back on her fingers, meeting her thrusts with the same fervor. the feeling of her hand slapping against your cunt becomes too much. 
abby squeezes your throat slightly, reminding you of her words. “i know you’re close baby. tell me what you are and i’ll let you cum,” she urges, squeezing your throat once again. the loss of airflow was dizzying, only making your cunt tighten even more at the feeling. 
you crane your neck back to look at her, giving her the most desperate look that she now finds herself obsessed with. you struggle to find the words, itching to please abby by doing what you’re told. 
“nngh ‘m a dumb slut... jus’ for you,” you whimper, eyes rolling back as your orgasm rushes through your entire body. your legs shake, body writhing against the table as your whole body goes lax. abby ruthlessly fucks you through it, fucking you into overstimulation. 
after what feels like a lifetime, she slides her fingers out of your stretched hole, giving it a small smack just to watch you jump. she slides your panties back into place, letting the slick from your recent orgasm pool into them. 
you watch through hazy half-lidded eyes as she gathers her things, slipping them into her bag. she gets a piece of paper, scribbling something on it before placing it back on the table. 
she rounds the table back to you, giving you a chaste kiss on your spit-covered lips, leaving a trail when she pulls away. her face is calm with a hint of smugness as she caresses your cheek. “our next session is tomorrow at your place. maybe you’ll actually learn something then,” she smiles, adjusting her bag and walking away. 
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bearseulgs · 1 year ago
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[16:32]
tutor!reader x ni-ki
genre: fluff
wc: 347
warnings: cheesy flirting, reader jokingly calls Riki a "sleaze", statistics class
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You sat in the library, waiting for Riki to arrive. When he got there, you opened your bag and got your work out as he did the same. The table was quite covered with all the schoolwork you needed to cover with him. Leaving your bag open, you made sure to tell the boy, "If any of your belongings fall into my bag, feel free to grab them, no hesitations." He just nodded and you smiled at him.
About an hour had passed, half of your session over already, and Riki seemed to be in his own world as you moved on to the next topic. You rapped your pencil sharply on his textbook, startling him back to reality. He apologized sheepishly as you chuckled at his behavior. "What about this chapter is confusing you?" you asked, returning your attention to the statistics papers in front of you two.
"Ah... all of it together? I know some individual concepts, but once you put in all these words together, like discrete random variables and the Z-score, I get lost."
"Understandable. Let me get some notes out, I think I have an explanation written somewhere." You rifled through your bag, looking for the probability notes that had come out of your binder. Huffing in frustration, you dug further, unable to find them, when suddenly you felt a warm hand on yours. Looking up in surprise, you see Riki holding your hand in his own. "Is everything alright, Riki?"
He smiled, a bit abashed, and brought your hand up to the table. "You said if anything of mine fell into your bag, to grab it, no hesitations." He laughed as you withdrew your hands from his, bringing them up to your face to hide your embarrassment.
When you collected yourself enough to look at him, you mumbled, "Gosh, Riki, you're such a sleaze," not meaning a word you said.
He feigned confusion, innocently replying, "But I was just getting my things out to study statistics." As you rolled your eyes, he laughed again.
"Go back to your textbook, Riki."
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a/n: this was such a great concept in my head and i have no idea how to fully properly convey it in any human language i speak
© bearseulgs 2024
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altin-studies · 1 month ago
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eid vibes
Exam diaries II.III, dop 1/100
Sunday, 15 June, 2025
Next Exam: Sampling Distributions (18.06)
I am both relieved and terrified of my sophomore year finals this year. I need so much more marks for As. ADHD has been rampant lately. I totally lost all joy in studying. Pushing through anyways...
Today's to-do: (insha'Allah)
Pray all 5 prayers F D A M I
Discrete Probability Distributions: pmf, cdf, use, pgf, mgf, mean, variance
Continuous Probability Distributions: pdf, cdf, use, mgf, mean, variance
Generating functions: pgf,pmf,cgf
Distributions function technique of finding distribution of a random variable
Distribution of minimums and maximums
Nothing is easy except for what Allah makes easy and he can make all difficult easy đŸ€ČđŸœ
Hope you guys have a great time studying ~
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moonlight-n-moondust · 5 months ago
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day 73/100
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hey guys, im back
im still so tired rip
staying up overnight is just not it
but i was more productive than thurs
finishes gp essay
did 6/11 qns on discrete random variables for statistics
wasn't as productive as i hoped for tdy
will try to clear more backlog tmr
~2 hours 30 min on ypt~
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nohoperadio · 2 months ago
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now that i've remade my account, i can finally send asks again! this is my first one. i suppose i should actually ask something though so here's my question:
have you learned anything new about hands since you've started your hand collection? not necessarily like, hard scientific facts, new aesthetic information also counts
congrats on not being a ghost anymore! I'm pleased
idk if there's like discrete takeaways from the hand thing but it's definitely emphasized how much different people's hands differ from each other, I mean it's probably not quite the case that I could look at a random set of newly-taken hand pics and match them up with specific mutuals, but I could definitely narrow each one down significantly and I bet I'd guess a lot right. There's just a lot of different variables! In textures/shapes/proportions/colours etc, just a lot of ways to be a hand, you know! Glance through the tag and you'll see
I guess one specific piece of microtrivia was the thing about how resting your palm flat against a hard surface tends to make tendons invisible, which isn't something I would have ever noticed otherwise and I can tell it's the sort of little detail that if I was an artist I'd want to squirrel away and play with (sadly I'm not an artist, my senses only take and consume and are barren)
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yaldabaothadeez · 4 months ago
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So, putting some things I've been talking about recently together: this is what I understand is going on with Kelly bets and the whole "ergodicity economics" thing:
You have some system with randomness and a dynamic - what happens next depends on what went before. We'll model this as a discrete stochastic process X_n for n=1 , 2, ... but you could make it continuous instead. For Kelly, this dynamic was compound interest: X_n = X_{n-1} * [random variable]
Actually, you have a choice several such systems, and you want to know which one is the "best" in some sense - assume bigger values mean better - but obviously X_n is a random variable, it looks you have to make some decision about what to measure (e.g. a "utility function" in the sense of von Neuman-Morgernstern)
But wait! If you're really lucky, X_n might converge to some deterministic f(n) (converge in probability? almost surely?) and then you can just choose the option whose f(n) is growing fastest.
This absolves you of having to find a vNM utility function, as the fastest-growing option will always be the best, eventually, so you can stop thinking about probability (Assuming you care about what happens asymptotically, or at least after enough rounds that it's a good approximation)
The easy way to show X_n converges seems to be to find an invertible monotonic function u such that Y_n = u(X_n) is a sum of n i.i.d random variables (i.e. Y_n = Y_{n-1} + A, for the same random variable A each time), and then apply the law of large numbers: as n gets large Y_n -> nE(A) and so X_n -> u^-1(n E(A)) - so, assuming u is increasing, you pick the option with the biggest E(A).
(The continuous versions of this I've seen seem to want to map the variable into Brownian motion with a constant drift and variance)
This feels like overkill to me? We don't need i.i.d variables for the law of large numbers, just suitably bounded growth in the variance, and that's not the only way things can converge.
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blackorchid-whitelily · 7 months ago
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Welcome, weary traveler.
Admin info:
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silenceitis · 4 months ago
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An intuitive introduction to information theory, part 1
Motivation
Ok, let’s start with the basics; what exactly is information theory and why do we care about it?
As the name suggests, well, it’s the theory of information. More precisely, how do we “measure” how much “information” something has? What is the “best way” to represent this “information”? How can we communicate this “information”? How can we communicate information if there’s a chance some of it may be “lost” in transit?
As you may have guessed, this underpins many important technologies we all use today, such as digital communication and file compression.
All in all, it’s about developing a framework that allows us to quantify “uncertainty” and optimise communication.
Important ideas
Ok, so I’ve said the word “information” a lot. But what does this actually mean? Let’s think about the concept of surprise. If I told you “the sun will rise tomorrow”, you wouldn’t be very surprised would you? Or in other words, this statement wouldn’t give you much information. You are already pretty certain that the sun will indeed rise tomorrow. But what if I told you “your exam scheduled for 2 months time has been moved to tomorrow”? That would be pretty damn surprising right? Or, you could say it’s a very informative statement.
So, we can see that there is some relationship between the idea of surprise and the idea of information. Let’s try and describe this intuition a bit more precisely.
Clearly, how surprised we are about something is related to the probability of that thing happening. If I had a biased coin that had a 99% chance to land on heads, and 1% to land on tails, you’d be very surprised if I flipped a tails, and not very surprised at all if I flipped a heads. So surprise decreases as the probability of some event happening increases.
Further, if we have two completely independent things happening, it would make sense for our surprise to add. Going back to the same biased coin, suppose I flip it, and you feel whatever surprise you do about the result. If I flipped another biased coin at the same time, you would feel a certain amount of surprise about this too. But the amount of surprise you felt about the result of one of the flips wouldn’t change the amount of surprise you felt about the other, the two coins don’t affect each other do they?
And finally, if an event becomes slightly more likely, or slightly less likely, we would expect our amount of surprise to only change slightly too right? That is, the amount of surprise we feel about an event should depend continuously on the probability of the event.
If we denote s(A) to be our surprise for the event A, it turns out that S(A) = -log(p) satisfies all these properties, where p is the probability of A happening. Not only that, but this is actually the only function that satisfies these properties (up to our choice of base for the logarithm).
Ok, so we now have a sense of what we mean by surprise, and hence information, and an explicit function to calculate a numerical value representing this. But notice how to make use of this, we need to actually know what the outcome is. E.g. if I have a biased coin, you can’t tell me anything until I flip the coin and you interpret the result. But what if I have a coin and I haven’t flipped it yet? Based on the probabilities, we can work out how surprised we *expect* to be. Or in other words, on average, how surprised you will be.
This is exactly the concept of entropy!
Now, I’ve talked a lot about coin flips. You’re probably getting a bit bored of this. So let’s formalise things slightly. Suppose we now have a discrete random variable X, taking values in a set χ. As you may recall from any probability classes you’ve taken, the expected value of X can be calculated as
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And we can calculate the expected value of a function of X as
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Now, surprise is just a function of our random variable X, so we can use this formula to work out what entropy is! We will write entropy as H(X). Again, H(X) is just a value representing, on average, how surprised we are when we observe an event from X.
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From this formula, we can say a few things about entropy.
Firstly, it it is always non-negative, which we expect; being “negatively surprised” about something doesn’t really make any sense right? So why would our average surprise ever be negative?
Furthermore, entropy is only zero when one of the probabilities in the sum is 1 (remember log(1) is zero, and probabilities sum to 1). Again, this makes sense. Our average surprise would only ever be zero if we were certain of what the outcome would be.
Ok great, we now know, given a random variable X, on average how surprised we will be. But what if we don’t actually know what the random variable X is? This happens all the time! In real life, we may be able to observe a random process, and hypothesise what we think the distribution is, but we may not know for certain. For instance, suppose I had a biased coin (yep, sorry, back to the coin again
), but I don’t tell you what the probabilities of landing on heads or tails are. You may take the coin, do a whole bunch of flips, and estimate the probabilities from your results. But your estimate is unlikely to be the true answer. We want to find a way to measure “how costly” it would be to use your estimate, rather than the true distribution. Or in other words, how much information we lose (or indeed surprise we gain) by using your estimate.
This leads us to the idea of divergence, a way of measuring how different two probability distributions are, by looking at the differences in entropy (aka surprise, aka information!) between the two.
Suppose we have a random variable with probability mass function p, on the set χ. But we want to encode X with a different pmf, q, also on χ. Then the average surprise when taking values from X and encoding with p (aka using p in our function S(A)) is just the entropy of X, H(X), as we discussed before.
And the average surprise when taking values from X but encoding with q (again, just using q in our surprise function) is
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Notice how this is not quite the entropy of X, although it does look very similar. Our surprise has changed, as surprise is based on our “beliefs” of what the values of X will be, that is, it is based on our choice of encoding. But our “beliefs” do not change the actual probabilities of X taking particular values.
Hence, our divergence, written D(p || q), is just
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However, the concept of divergence is not just used to compare how our guess for the distribution, q, compares with the true distribution, p. It can actually be used to compare how different *any* two distributions are, as long as they are both on the same set of values. More precisely, D(p || q) measures how different the two distributions are, *from the perspective of p*. I.e. we draw values from p, and measure how much more surprising that sample would be, on average, if it had come from q instead.
We do precisely the same thing as before, except instead of entropy, we have the slightly modified surprise function, as we did for q. Note that at no point have we made an assumption on what the actual distribution of X is here (which is why the sum is not the entropy of X, even if it looks very similar).
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Now, as you may have noticed, what I said before hints at some asymmetry in the definition of divergence, and this is indeed the case. In general, D(p || q) is not equal to D(q || p)! If we think about the meanings of these two quantities, this makes sense. D(p || q) represents how “inefficient” it is to pretend that data from p came from q. Whereas D(q || p) represents how “inefficient” it is to pretend that data from q actually came from p. Intuitively, these are not the same thing.
Additionally, you may have noticed that the divergence can take the value infinity, when for some value of x, p is greater than zero, but q is zero. Intuitively, this means that q, the distribution you’re pretending the data came from, says that a certain event is impossible, but p says that this event *does* occur.
Ok, so now we’ve seen how we can “measure” how different one distribution is from another, which can be useful when we want to model an unknown distribution. But what if we don’t try to estimate our unknown distribution directly? Instead, we may be able to observe another random variable, and hypothesise that it may be able to tell us something about our unknown distribution.
This leads us to the idea of mutual information. How much does knowing Y tell us about X? Or in other words, how much is our uncertainty in X reduced by, if we also observe Y?
So really, we’re looking for a way of determining how much dependence there is between the two variables. If X and Y were independent, then knowing something about Y wouldn’t reduce our uncertainty in X. So we can measure how different the joint distribution of X and Y is, from the joint distribution if they were independent! This will give us exactly what we’re looking for; how “far” the joint distribution of X and Y is from being independent.
We can use our notion of divergence to calculate this! As you may recall, if X and Y are independent random variables, the joint pmf of X and Y is simply the pmf of X multiplied by the pmf of Y. So we just plug this into our divergence function, as below, writing I(X; Y) for the mutual information:
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And with some rearranging and use of our previous definitions, we can get the following alternate forms for mutual information:
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All of these definitions make it clear to us that mutual information is in fact symmetric, I.e. I(X;Y) = I(Y;X).
We can also see that mutual information is zero if and only if X and Y are independent, and that it is non-negative.
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esme-studies · 6 months ago
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study session: statistics, discrete random variable
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annes-room · 1 year ago
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đŸ–Šïž May 21, 2024
discrete random variables my beloved 💖
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codingprolab · 3 days ago
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ISTA 421/521 – Homework 3
1 1. [2 points] Adapted from Exercise 2.3 of FCMA p.90: Let Y be a random variable that can take any positive integer value. The likelihood of these outcomes is given by the Poisson pmf (probability mass function): p(y) = λ y y! e −λ (1) By using the fact that for a discrete random variable the pmf gives the probabilities of the individual events occurring and the probabilities are additive
 (a)

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jarviscodinghub · 17 days ago
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CS 4650/7650 Homework 1 Solved
2. The entropy of a discrete random variable X is defined as (use base e for all log operations unless specified otherwise): H(X) = − X x∈X P(x) log P(x) (a) Compute the entropy of the distribution P(x) = Multinoulli([0.2, 0.3, 0.5]). [3 pts] (b) Compute the entropy of the uniform distribution P(x) = 1 m ∀x ∈ [1, m]. [3 pts] (c) Consider the entropy of the joint distribution P(X, Y): H(X, Y ) =

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lolboostsu · 2 months ago
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