Find Var(x) of joint probability distribution

[Guavajuice]

Hi!

I was doing this question on statistics, where I was given the joint probability distribution

f(x,y) = (9/16) (1/4^(x+y))

Where x and y were discrete random varaibles
x = 0,1,2,... And y = 0,1,2,...

I was then tasked to find E(X), E(Y) and Var(X) and Var(Y)

I managed to find E(X) and E(Y) =1/3, by finding sum to infinity for Geometric Progression for the y-term, and using an arithemtic-geometric progression sum to infinity to find the x-term.

However, when I was trying to find var(x)

I was stuck at this summation

Summation of x term , from x= 0 to x= infinity, of the function (x^2)/4^x

I'm not sure what kind of series this is, because, it doesn't seem to be an AP/GP/ or AGP, since the numerator is a square of X, while the denominator is an exponential function.

Any

 

[Steven G]

Did you look at the first few terms? What have you tried? To receive help we need to know where you are stuck. Yes, there are series that are not AP or GP! If you are in a probability course you must have taken calculus so you have seen many different type of series. Does your series even converge? Why/why not? You have to be able to try something. Please post back showing some work and then you'll receive a hint.

 

[Steven G]

Maybe use some software? Is that allowed?

 

[Guavajuice]

Did you look at the first few terms? What have you tried? To receive help we need to know where you are stuck. Yes, there are series that are not AP or GP! If you are in a probability course you must have taken calculus so you have seen many different type of series. Does your series even converge? Why/why not? You have to be able to try something. Please post back showing some work and then you'll receive a hint.

hi[MATH][ATTACH type="full"]20240[/ATTACH] hi so for context, i am beginning university in a month so i decided to buy the statistics textbook to study earlier, so i was stuck with this question as i had only learned AP and GP in school. When solving for E(X), I had come across an AGP, so i learnt the derivation and solved it. However for Var(X), i realised this x^2/4^x did not follow an AGP/AP/GP, so i tried to derive Sn for this new series, but it seemed really complicated and felt like it was wrong (brown image). I tried to plug in the sum to my answer (white paper), but got the answer as 11/12, which was wrong. Any idea where I can learn more about different series etc? Im sure this series converges as i plugged it into my graphic calculator and it approaches an asymptote. Thanks for the help! Also, software as in using a graphic calculator? I havent learnt how to use python etc for math applications so probably an answer without software is preferable. [/MATH]

 

[Steven G]

Summation of from x= 0 to x= infinity of (x^2)/4^x - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

 

[Guavajuice]

Summation of from x= 0 to x= infinity of (x^2)/4^x - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

www.wolframalpha.com

Hi, thanks for the link,

However I am looking for an explanation on how I can solve the sum to infinity manually, by breaking down the terms etc into different series etc.

If you have any idea on how I can do that, rather than just plucking the equation into the software it would greatly be appreciated!