What is the expectation? $E\left[\left(\frac{q}{p}\right)^{X_n}\right]$?

Question

i'm just trying to figure this out.

$$P(X_n = 1) = p \mbox{ and } P(X_n =−1)=q \\ \mbox{for each }n∈N(p,q∈(0,1),p+q=1,p>q)$$

what is $$ E\left[\left(\frac{q}{p}\right)^{X_n}\right] $$

This is part of a bigger martingale question, it is just this part I do not understand, the expectation given was $(p+q)$ which equal to 1. How do I do this part?

Thanks for your help. (psttt : this is also my first question on stackexchange ;))

Answer 1

Answer from Did:

$E[(\frac{q}{p})^{X_n}]=(\frac{q}{p})^1P(X_n‌=1)+(\frac{q}{p})^{−1}P(X_n=−1)=\frac{q}{p}p+\frac{p}{q}q=p+q=1$