Let $n$ be a positive integer. Consider a random walk with $S_n=1+\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively.
I would like to calculate $E(S_n)$. My attempt:
$$E(S_n)=E\left(1+\sum^n_{i=1}X_i\right)=1+\sum^n_{i=1}E(X_i)$$ $$=1+n[(-1)1/9+(0)1/9+(2)7/9]=1+(n)\frac{13}{9}$$
Would this be correct?
Best Answer
It is correct. The only thing you use here is linearity of expectation.
Remark: Furthermore, there is no need for independence here.