Question

Let X and Y be independent; each is uniformly distributed on [0, 1]. Let Z = X + Y. Find:

E[Z|X]. Your answer should be a function of x.

Answer 1

As we are given that X and Y are independent, therefore the value of Y will not depend on any value for X.

Now we are given here that Z = X + Y,

For a given value of X = x, the value of Z is computed as:
Z = Y + x, where x is a constant as it is given.

Also, we know here that:

Therefore the distribution of X would be given as:

Therefore now that we know the conditional X = x, the distribution of Z is uniform, the conditional expected value of Z given X = x, is computed here as:

Therefore x + 0.5 is the required expected value here.