Question 1: Expectation and variance - discrete Let Y have a uniform discrete distribution. In other words, P(Y = y) = \frac{1}{b-a} for Y taking on integer values in the range a < Y \le b (note the strict inequality).
a. First, consider the special case of the random variable X, which is also discrete uniform, but let a = 0 and b = n, so that P(X = x) = 1/n for the first n integers. Find the mean and variance of X. You may find the following facts handy:
$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$
b. Next find a formula for the mean and variance for Y. Use linearity of expectation.