yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Example: Transforming a discrete random variable | Random variables | AP Statistics | Khan Academy


3m read
·Nov 11, 2024

Anush is playing a carnival game that involves shooting two free throws. The table below displays the probability distribution of ( x ), the number of shots that Anush makes in a set of two attempts, along with some summary statistics.

So here's the random variable ( x ): it's a discrete random variable; it only takes on a finite number of values. Sometimes people say it takes on a countable number of values, but we see he can either make 0 free throws, 1, or 2 of the two. The probability that he makes zero is here, one is here, and two is here. They also give us the mean of ( x ) and the standard deviation of ( x ).

Then they tell us if the game costs Anush fifteen dollars to play and he wins ten dollars per shot he makes, what are the mean and standard deviation of his net gain from playing the game ( n )?

Right, so let's define a new random variable ( n ), which is equal to his net gain. Net gain can be defined in terms of ( x ). What is his net gain going to be? Well, let's see: ( n ) is going to be equal to 10 times however many shots he makes. So it's going to be ( 10 \times x ), and then no matter what, he has to pay 15 dollars to play, minus 15.

In fact, we could set up a little table here for the probability distribution of ( n ). So let me make it right over here. I'll make it look just like this one. ( n ) is equal to net gain, and here we'll have the probability of ( n ). There's three outcomes here.

The outcome that corresponds to him making 0 shots: well, that would be ( 10 \times 0 - 15 ); that would be a net gain of negative 15. It would have the same probability ( 0.16 ).

When he makes one shot, the net gain is going to be ( 10 \times 1 - 15 ), which is negative 5. But it's going to have the same probability; he has a 48% chance of making one shot, and so it's a 48% chance of losing 5.

Last but not least, when ( x ) is 2, his net gain is going to be positive 5, ( +5 ). And so this is a 0.36 chance.

So what they want us to figure out are what the mean and standard deviation of his net gain are.

First, let’s figure out the mean of ( n ). Well, if you scale a random variable, the corresponding mean is going to be scaled by the same amount. And if you shift a random variable, the corresponding mean is going to be shifted by the same amount.

So the mean of ( n ) is going to be ( 10 \times \text{mean of } x - 15 ), which is equal to ( 10 \times 1.2 - 15 ). This is ( 1.2 ), so it is 12 minus 15, which is equal to negative 3.

Now the standard deviation of ( n ) is going to be slightly different. For the standard deviation, scaling matters. If you scale a random variable by a certain value, you would also scale the standard deviation by the same value.

So this is going to be equal to ( 10 \times \text{standard deviation of } x ). Now you might say, what about the shift over here? Well, the shift should not affect the spread of the random variable. If you're scaling the random variable, your spread should grow by the amount that you're scaling it. But by shifting it, it doesn't affect how much you disperse from the mean.

So, standard deviation is only affected by the scaling but not by the shifting here. So this is going to be ( 10 \times 0.69 ), which is going to be approximately equal to 6.9.

So this is our new distribution for our net gain, this is the mean of our net gain, and this is roughly the standard deviation of our net gain.

More Articles

View All
Marques Brownlee on Building an Audience and Other Advice for Creators
All right Marques Brownlee, how’s it going? Good, how are you? Doing well! So I’m curious, I’ve followed your channel for a while, but I definitely did not follow it in the beginning when you were reviewing software on your laptop. You’ve been doing it …
Strategies for dividing multiples of 10, 100 and 1000
We’re going to do in this video is get some practice doing division with numbers that are multiples of 10, 100, 1000, things like that. So, let’s say we wanted to compute what 2400 divided by 30 is. Pause this video and see if you can calculate it using w…
3 tips for finding a job on YC's Work at a Startup
[Music] [Applause] [Music] [Applause] Thanks for joining Y Combinator’s Work at a Startup and welcome to the YSE network. I’m Ryan and I’m here to help you find your dream job. Y Combinator is an accelerator that has invested in companies like Coinbase, …
Homeroom with Sal & Vas Narasimhan - Tuesday, August 17
Hi everyone, Sal Khan here. Welcome to Homeroom with Sal. We have a very exciting show today. After a bit of a hiatus, we haven’t done a live stream in a little while, but we have Vas Narasimhan, who is the CEO of Novartis. We had him on last year at the …
Rethinking Our Relationship With Water | National Geographic
It’s hard to believe the world could ever run out of fresh water. Even though we live on a blue planet, only about three percent of Earth’s water is fresh. Of that, only one percent can be used as drinking water, and that is threatened by climate change a…
Covalent bonds | Molecular and ionic compound structure and properties | AP Chemistry | Khan Academy
In a previous video, we introduced ourselves to the idea of bonds and the idea of ionic bonds, where one atom essentially is able to take electrons from another atom. But then, because one becomes positively charged and the other becomes negatively charge…