I have this question from a previous exam:
At any given time, a particle can jump to the left, to the right, or stay in its position on a horizontal axis, each possibility with equal probability. Let $S_n$ be the position of the particle after $n$ steps. Note that the initial position of the particle is $0$.
I'm first asked to write $S_n$ as a sum of $n$ independent random variables that represent the displacements at different steps.
I wrote $S_n$ = $\sum_{i=1}^{n} X_i$ where $X_i = \left\{ \begin{aligned} -1 \\ 0 \\ +1 \end{aligned} \right.$
Is my $S_n$ correct? And the question that follows asks us to determine $Var(S_{2500})$, so how do we do that?