Ex:           Given X~u(0, 1), (i.e., X is uniformly distributed from 0 to 1), and Y = 5X + 1, find the following values:

a)      μY.

b)      .

Sol'n: a)  The mean of Y is given by a standard formula:

Substituting values given in the problem, we have the following result:

The mean value of X is 1/2 since it has a uniform distribution on (0, 1). Making this substitution gives our answer:

             b) The covariance, , is defined in terms of E(XY):

If we substitute Y = 5X + 1, we are left with only expected values involving x:

Using the result from (a), the second term simplifies as follows:

The first term becomes a sum:

To find the expected value of X2, we consider the variance of X. For a uniform distribution on (0, 1), the variance is 1/12.

Solving for E(X2), we have the following:

Substituting this result in our earlier equation, we have the information needed to complete the calculation of covariance:

and