Please can get aid on this

Question

A discrete random variable has probability mass function,P(X=n)= (1/2)ⁿ

^{let Y= {1, For x is even}

^{-1, For x is odd}}

^{find the expected value of Y; (E[y])}

Answer 1

Lydia,

I can sketch this out for you. First off, let's verify that this is a legal PMF, as this will help us both with information that was missing from the problem and provide us with helpful techniques for the rest of it. To do that we will sum it over all n and verify that the sum equals 1. Note that the PMF you are given forms a geometric series when summed over all n. If you remember, the sum of an infinite geometric series has the formula a1 / (1-r), where r is the number you multiply each term by to get the next term and a1 is the value of the first term. So here r=1/2, so the sum of the PMF over n is equal to 2a1, where a1 is the first term. Since it must equal 1, we know that a1=1/2, so this distribution starts at n=1, not n=0. (I'm assuming you forgot to mention that part, or it was mistakenly omitted by the prof.)

To find E[Y], you have to multiply the PMF by the value of Y for the given n and then sum up. When you do this, you'll get an alternating series which is also a geometric series. Therefore, you can use the same formula for the sum of the geometric series to calculate the expectation value of Y, although of course you will get a different answer since r and a1 are both different!

Hope that helps, Raphael