Let X and Y be zero-mean independent random variables. Which one of the following statements is correct? Hint: You can take for granted the intuitive fact that E[X/X=x] = x

Question

a) E[X+Y/X] = 0
b) E[X+Y/X] = x
c) E[X+Y/X] = X
d) E[X+Y/X] = X+Y

Answer 1

c) E[X+Y/X] = X

Answer 2

Why?

Answer 3

To find the conditional expectation E[X+Y/X], we condition on the value of X. Since X is a zero-mean random variable, when we condition on X, it takes on a fixed value x.

So, we can write E[X+Y/X] as E[X+Y | X=x].

By linearity of expectation, we can split this into E[X | X=x] + E[Y | X=x].

Since E[X/X=x] = x, we have E[X | X=x] = x.

Since X and Y are independent, E[Y | X=x] = E[Y] = 0, since Y is also a zero-mean random variable.

Therefore, E[X+Y | X=x] = x+0 = x.

So, the correct statement is c) E[X+Y/X] = X.