Let $X_1$, $X_2$, ... be a sequence of i.i.d random variables such that $P(X_1 = 2) = .4$, $P(X_1 = 1) = .2$, $P(X_1 = 0) = .4$. Calculate $E[X_1]$, standard deviation of $X_1$. And calculate approximately: $P(15 \leq X_1 +\dots + X_{25} \le 30)$.
I got the $E(X_1) = 1$ and the standard deviation to be the square root of 1.8, but how can I get the last part? My thinking was let $Z = X_1 + X_2 +\dots + X_{25}$ so then we will have $E[Z] = E[n X_1] = n \cdot 1 = 25 \cdot 1 = 25$. Am I on the right track?