Probability problems day 13

A generating function is a clothesline on which we hang up a sequence of numbers for display.

— Herb Wilf, generatingfunctionology (one of the best mathematics books ever written!)

1. Let \(Z\) be a standard normal. Compute \(E[Z^{2^{2^{100}} - 1}]\).

2. 1. Find the \(n\)th moment of \(U \sim \text{Unif}(0, 1)\).
   2. Find the \(n\)th moment about the mean of \(U\).
   3. Find the mgf of \(U\).
3. 1. Find a formula for the skewness of \(X\) in terms of its first three moments.
   2. Compute the skewness of a Geometric(1/2) variable. (Hopefully I computed the first three moments for you. If I didn’t, they are \(E[X] = 2\), \(E[X^2] = 6\), and \(E[X^3] = 26\).)

4. Compute the skewness of \(X \sim \text{binomial}(n, 1/2)\) in terms of \(n\). Hint. Try to repeat (3) with the binomial distribution.