Segment 7. Central Tendency and Moments

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To Calculate

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 Failed to parse (unknown error): p(x) , with a maximum at Failed to parse (unknown error): x=0 , whose third moment Failed to parse (unknown error): M_3 exists, but whose fourth moment Failed to parse (unknown error): 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 Failed to parse (unknown error): p(x) is a known probability distribution. But what if you know Failed to parse (unknown error): 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 Failed to parse (unknown error): I_8 , the eighth semi-invariant?

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