Expected value of random walk

Question

Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$.

I would like to calculate $E(S_n)$. My attempt:

$$E(S_n)=S_0+E\left(\sum^n_{i=1}X_i\right)=1+\sum^n_{i=1}E(X_i)=1+n\bigg{[}(-1)1/9+(0)1/9+(2)7/9\bigg{]}=\\1+n\frac{13}{9}$$

Would this be correct?

Answer 1

This is correct. The first equality stems from linearity of expectation, and the second equality is LOTUS: $\mathbb E[X]=\sum_{a\in\mathscr A} a \cdot \mathbb P[X=a]$.