Question

Let y1 and y2 have a joint density function given by f(y2, y4) = 6(1 - y4), for 0 ≤ y2 ≤ y4 ≤ (a) Find E(y1 | y2 = y2).

(b) Use your answer from (a) to find E(y1)

Answer 1

To find E(y1 | y2 = y2), we need to calculate the conditional expectation of y1 given y2 = y2. To find E(y1), we can use the law of total expectation.

Step-by-step explanation:

To find E(y1 | y2 = y2), we need to calculate the conditional expectation of y1 given y2 = y2. The conditional expectation is defined as the expected value of y1 given the value of y2. In this case, since y2 = y2, we can simply write E(y1 | y2 = y2) = E(y1 | y2).

To find E(y1), we can use the law of total expectation, which states that E(y1) = E[E(y1 | y2)]. Since we have already found E(y1 | y2 = y2) to be E(y1 | y2), we can substitute it into the equation to get E(y1) = E[E(y1 | y2)].