Help with Expectations.

1) Suppose we have a bucket containing marbles in the following colors and quantities: 1 red, 2 blue, 3 yellow, 4 green, and 5 purple. Draw 5 marbles from the bucket, with replacement, and let X be the number of colors that you draw exactly once. Find E[X]. Give your answer as a single fraction or a decimal.

I understand that the first derivative of the generating function evaluated at Z=1 would give the answer to the question. But it seems like coming up with the probabilities for the different values of X will not be easy. I'm wondering if there is an easier method to go about this problem.

2) Suppose GX(z) = (z^2)e^(2z-2). Compute:
(a) E[X].
(b) Var[X].
(c) P[X = 3].
(d) P[X = n].

For part (a), I just differentiated the function and evaluated @ z=1 which gave me an answer of E[X] = 4. Could anyone confirm is this is correct. I am not sure how to find E[X^2) to compute Var[X]. Also for part C and D, I am unsure how to get started with them.

Any help would be appreciated. Thanks.

It's a Moment Generating Function.

G' Produces the first moment
G" produces the second moment
etc.

The variance should be p;g-hat to you if you are talking about moment generating functions. Var(x) = E[X^2] - (E[X])^2 -- Ringing any bells?