# Random variables and sums of $k$-sided dice

Consider fair $k$-sided dice with the numbers $1$ through $k$ on their faces.
a. Roll one die. Let the RV $X$ be the number on one die. Compute $E[X]$ and $V[X]$.
b. Roll $n$ dice. Let the RV $Y$ be the sum of numbers on the dice. Compute $E[Y]$.
c. Roll two dice and let the random variable $Z$ be the second die subtracted from the ﬁrst. What is $E[Z]$?

-

Hints: For (a), do you know formulas for $\sum_{i=1}^k i$ and $\sum_{i=1}^k i^2$?
For (b) and (c), use the linearity property of expected value: for any random variables $X$ and $Y$ whose expected values exist and any constants $a$ and $b$, $E[aX + bY] = a E[X] + b E[Y]$.