Review

• More on "hot hands" and sports streaks:
  • Stephen Jay Gould's marvelous essay on Joe Dimaggio's streak
  • Alan Reifman's page on the Hot Hand in Sports
• Red Sox probability problem: what's the probability that they will win, given that they are 2-0 in the series and assuming that the teams are equally matched?
• Standard Error of the mean:

  σ_x-bar = σ/sqrt(n)

The Central Limit Theorem

Amazingly enough:

The distributions of sums (and therefore means) is increasingly Gaussian

We'll explore the following model

CLT2

Limitations of the CLT:

• The samples must be independent
• The factors must be additive in effect. That is, if some measurement is the result of many independent factors, we might want to invoke the CLT and assume it is Gaussian, but not if the factors have a synergistic effect, say. Example: the area of a rectangle is not an additive result of the width and height.
• The CLT says that the distribution approaches the Gaussian, but it doesn't say how fast. In practice, it's amazingly fast, but if you really need high accuracy, you wouldn't want to rely on the CLT.