Week 11 Preparation Guide

Reading

Required:

• Textbook Chapters 18 and 19

Recommended:

• Pitman Section 5.3 (pages 361-363 have a lovely trigonometric proof of why sums of independent normals are normal; we’ll be using that diagram in a couple of weeks).

• Pitman Section 5.4 through page 376. Try not to be intimidated by the notation in Example 2. The calculation works like magic and is much easier than it looks.

Pitman’s text doesn’t cover mgf’s. Between lecture and homework, Prob140 covers pretty much all the mgf’s that are both interesting and can be reasonably computed at this level.

Practice Problems

Pitman x.y.z means Exercise z of Section x.y and x.rev.z means Exercise z of the Review Exercises at the end of Chapter x.

• Pitman 5.3.3, 5.3.4, 5.3.9, 5.3.15 (parts d-f are on the next page; most of this problem was done in class; for part (b) use the recursion for the gamma function that you derived in HW 10 and just write the terms for n = 3, 5, and 7; in part (f) ignore skewness)

• 5.rev.16 (use what you know about sums of squares of standard normals), 5.rev.22 (crucial: if $X$ is normal $(0, \sigma^2)$ then $X = \sigma Z$ for a standard normal $Z$)

• Compute the mgf of the uniform distribution on (0, 1). To see that you got the right answer, use it to calculate the first two moments and check that they agree with the corresponding integrals.

• Calculate the mgf of the geometric (p) distribution on ${1, 2, 3, …}$. That’s the distribution of the number of tosses till the first head. Use it to find the expectation; that will indicate whether you got the answer right.

Discussion Section

• Warm-up (5.3.3, 5.3.15c), 5.3.4, 5.rev.22, mgfs of the uniform and the geometric