Suppose we generate a random variable X in the following way. Forst we flip a fair coin. If the coin is heads, take X to have a Uniform(0, 1). If the coin is tails, take X to have a Uniform(3, 4). (a) Find the mean of X. (b) Find the standard deviation of X.

(a) The mean of X is 2.

(b) The variance X is .

Step-by-step explanation:

The random variable X defined as follows:

X = heads, then X follows Uniform (0, 1).

X = tails, then X follows Uniform (3, 4).

(a)

The expected value of a Uniform random variable U (a, b) is:

Compute the expected value of X = heads as follows:

Compute the expected value of X = tails as follows:

Compute the mean of X as follows:

Thus, the mean of X is 2.

(b)

Compute the variance of X as follows:

Thus, the variance X is .