Let $X$ be a random variable representing the roll of one die.
Then,
$E(X) = \dfrac{1+2+3+4+5+6}{6} = 3.5$
$E(X^2) = \dfrac{1^2+2^2+3^2+4^2+5^2+6^2}{6} = \dfrac{91}{6}$
$Var(X) = E(X^2)-E(X)^2 = \dfrac{91}{6} - \dfrac{49}{4} = \dfrac{70}{24}$
$\sigma = \sqrt{Var(X)} = \sqrt{\dfrac{70}{24}} \approx 1.71$
Let $Y$ be a random variable representing sum of the rolls of $N$ dice and $S$ representing the set of possible outcomes.
How can I find $P\{ A \leq Y \leq B\}$ for some $A,B \in S, A < B$?