Eleisha's Segment 7: Central Tendency and Moments
1. Prove the result of slide 3 the "mechanical way" by setting the derivative of something equal to zero, and solving.
2. Give an example of a function p(x), with a maximum at x=0, whose third moment M_3 exists, but whose fourth moment M_4 doesn't exist.
3. List some good and bad things about using the median instead of the mean for summarizing a distribution's central value.
To Think About:
1. This segment assumed that p(x) is a known probability distribution. But what if you know p(x) only experimentally. That is, you can draw random values of x from the distribution. How would you estimate its moments?
2. High moments (e.g., 4 or higher) are algebraically pretty, but they are rarely useful because they are very hard to measure accurately in experimental data. Why is this true?
3. Even knowing that it is useless, how would you find the formula for I_8, the eighth semi-invariant?