Question

Conditional expectation. A fair die is continually rolled until an even number has appeared on 10 distinct (not necessarily consecutive) rolls. Let X, denote the number of rolls that land on size ili = 1,2,3,4,5,6). (a) Find E[X2]. (b) Find E[X1]. (Hint: think about the number of ones between the j - 1st and jth even values). Hint: condition on whether a 1 appeared before the first even number of after.

Answer 1

To find E[X^2], we calculate the expected value of X^2. The probability of getting an even number on the first roll is calculated, and then each subsequent roll is considered. The expected value of X^2 is determined using a series of probabilities.

Step-by-step explanation:

In order to find E[X^2], we need to calculate the expected value of X^2. Let's first consider the probability of getting an even number on the first roll. The probability of this happening is 3/6 (since there are three even numbers on the die). In this case, X^2 = 0 (since no rolls will land on a one).

If we don't get an even number on the first roll, we have another probability of getting an even number on the second roll. The probability decreases because now we have only two even numbers left out of five possible outcomes. In this case, X^2 = 1 (since we have one roll landing on a one).

We continue this pattern until we find the expected value of X^2. The final value is: E[X^2] = (0*(3/6)^0 + 1*(3/6)^1 + 2*(3/6)^2 + ... + 9*(3/6)^9 + 10*(3/6)^10).