Question

Fair diced, which is unbiased. Each throw is independent.

Step 1. You roll a six-sided die. Let X be the (random) number that you obtain. Step 2. You roll X six-sided dice. Let Y be the total number (sum) that you obtain from these X dice. Find E[Y], rounded to nearest .xx.

Answer 1

A six-sided fair die is rolled and the outcome is noted in a random variable X. So, X takes values from 1 to 6 with equal probability of 1/6. Now, given X = r, I other six-sided fair dice are rolled and their sum is denoted by Y. If Y1, ... ,Y, denote the outcome of different rolls, then each Y; has the same distribution as X and also Y = Y1 +...+Y.. We know that, E(X)= (1+2+3+4+5+6) = 3.5. And from above we can say, E(Y|X = x) = E(Y1 + +Yx|X = x) = E(Y1 + · +Y) = E(Y1) + + E(Y) =r. E(Y) = 3.5.2 (since Yi's are identical in distribution) (Y1 has same distribution as X and E(X) = 3.5) Then using Law of Total Expectation, we get E(Y) = E(E(YX = 1)) = E(3.5X) = 3.5E(X) = 3.5 * 3.5 = 12.25.