# Segment 7. Central Tendency and Moments

## Contents

#### Watch this segment

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Links to the slides: PDF file or PowerPoint file

### Problems

#### 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 $\displaystyle p(x)$ , with a maximum at $\displaystyle x=0$ , whose third moment $\displaystyle M_3$ exists, but whose fourth moment $\displaystyle M_4$ 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.

1. This segment assumed that $\displaystyle p(x)$ is a known probability distribution. But what if you know $\displaystyle p(x)$ only experimentally. That is, you can draw random values of x from the distribution. How would you estimate its moments?
3. Even knowing that it is useless, how would you find the formula for $\displaystyle I_8$ , the eighth semi-invariant?