Question

Let X be the number of heads in two tosses of a fair coin. Suppose a fair die is rolled X+1 times after the value of X is determined from the coin tosses. Let Y denote the total of the face values of the X+1 rolls of the die. Find E[Y | X = x] and V[Y | X = x] as expressions involving x. Use these conditional expected values to find E[Y] and V[Y].

Answer 1

The expected face value of one roll of a die=

                                                                

The expected in one roll of a die=

                                                                

Thus the variance is:

Let denote the face value of the die in the i th throw. Note Yi's are independent.

Then,

                       

                     

                      ,    because the 's are independent.

So,

Now, we will use these to calculate E[Y] and V[Y].

First, we know,    , so

Thus,

Adding these up,