Question

A discrete random variable has pmf P(X=n)=(1/2)^n. Let Y={1, if x is even and -1 if x is odd

Find the expected value of Y.

Answer 1

$P(X=n)=(1/2)^n$

$Y=\begin{cases} 1, \ \text{if x is even} \\ -1, \ \text{if x is odd} \end{cases}$

$P(Y=1)=\sum \limits_{n=1}^{\infin}P(X=2n)= \sum \limits_{n=1}^{\infin} (1/2)^{2n}= \sum \limits_{n=1}^{\infin} (1/4)^{n}=\frac{1/4}{1-1/4}=1/3$

$P(Y=-1)=1-P(Y=1)=1-1/3=2/3$

$E[Y]=1\times P(Y=1)+(-1)\times P(Y=-1)=1/3-2/3=-1/3$

Answer: the expected value of $Y$ is -1/3.