Consider the probability distribution of a random variable x

Consider the probability distribution of a random variable x. Is the expected value of the distribution necessarily one of the possible values of x? Explain.

Solution

No.

For example, let x = the number of heads in a single coin toss. Thus, the expected number of heads is 1/2.

However, x can never be 1/2, but rather can only be 1 or 0.

Thus, the expected value of the distribution is not necessarily one of the possible values of x.