i have the following Problem:
Let $M_n := (\frac{1}{p-1})^{S_n}$ where $S_n:= X_1 + X_2 + ... + X_n$ for $X_i \in \{-1,1\}$, $X_i$ iid with $P[X_i=1]=p$. Let $\textbf{F}_n:= \sigma(X_1,X_2,...,X_n)$ be the sigma algebra. Prove $M_n$ is a Martingale.
Clearly $M_n$ is bounded and adapted.
For the last property we get $E[M_{n+1}|F_n]=(\frac{1}{p-1})^{S_n}E[(\frac{1}{p-1})^{X_{n+1}}]$. Now the solution tells me the last expectation is equal to $p\frac{1-p}{p}+(1-p)\frac{1-p}{p}$.
My solution would $E[(\frac{1}{p-1})^{X_{n+1}}] = E[g(X_{n+1})] = g(1)P[X_{n+1}=1]+g(-1)P[X_{n+1}=-1] = (\frac{1}{p-1})^1p+(\frac{1}{p-1})^{-1}(1-p)$ .
Am I doing weird stuff or does the solution make no sense? I would be very thankful if somebody could check my sanity.