Arandom variable x that assumes the values x1, x2, . . , xk is called a discrete uniform random variable if its probability mass function is f(x) = 1 k for all of x1, x2, . . , xk and 0 otherwise. find the mean and variance of x.

Recall that for a random variable following a discrete distribution, the expectation of is given by



where is the PMF of . We have



So the expectation (mean) of the given uniformly distributed is



Without any more specific information, this is all we can say about the mean.

The variance of is defined by



In computing the second moment we run into the same issue as before - we can't find a "complete" result - but we do get



and so



We can expand a bit. Denoting the set , we can write



and so




but again, that's as much as we can say without any more specific information.