Slightly shorter homework this week since we only have one class. Although, there’s a lot to say about moments!

Moments

1. The Poisson distribution has a funny property: it’s easier to compute \(E[X(X - 1)(X - 2)]\) than \(E[X^3]\).

   1. Compute \(E[X(X - 1)]\) for \(X \sim \text{Poisson}(\lambda)\).
   2. Using (a), compute \(E[X^2]\). Hint. Do some algebra on (a).
   3. Compute \(E[X(X - 1)(X - 2)]\).
   4. Using (c), compute \(E[X^3]\).
   5. Conjecture and prove a pattern for \(E[X(X - 1)(X - 2) \cdots (X - m + 1)]\). Use this to figure out \(E[X^4]\).

2. The Gamma function is a special function defined by

\[\Gamma(z) = \int_0^\infty x^{z - 1} e^{-x} dx.\]

1. Use integration by parts to show that \(\Gamma(z + 1) = z \Gamma(z)\).
2. If \(n\) is a nonnegative integer, what is another way to write \(\Gamma(n + 1)\)? (As far as I know, the Gamma function was invented almost explicitly to do this.)
3. Show that, if \(X\) is exp(1), then \(E[X^n] = \Gamma(n + 1)\). What is another way to write this?

3. The kurtosis of a random variable is the fourth standardized moment:

\[\operatorname{kurt}(X) = E \left[ \left( \frac{X - \mu}{\sigma} \right)^4 \right]\]

This is one measure of how ‘fat’ the tails of a distribution are.

Compute the kurtosis of \(Z\), a standard normal.

4. A random variable is symmetric if \(X - E[X]\) and \(E[X] - X\) have the same distribution. Prove that the skew of every symmetric distribution is 0.

Moment generating functions

The following exercises ask you to compute the moment generating functions of some distributions that we have seen. Let me emphasize that literally all of these are available online. You should be able to derive them on your own, without help.

5. Find the moment generating function for the standard normal. Hint. Do you remember how to complete the square?

6. Find the moment generating function for the exponential distribution with parameter \(\lambda\).

7. Find the moment generating function for the uniform distribution on the unit interval.

8. Find the moment generating function for the Poisson distribution with parameter \(\lambda\).

9. Find the moment generating function for the Binomial distribution with parameters \(n\) and \(p\).

10. Let \(m(t)\) be the moment generating function of \(X\). Find a formula for \(\operatorname{Var}(X)\) in terms of \(m(0)\), \(m'(0)\), and \(m''(0)\). Hint. Remember that \(m(t)\) is a Taylor series of moments.