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Here are notes related to today’s class. Some have been linked previously.
Special probability distributions (summary notes) – was linked previously:
Distinguishing the special distributions:
Math2501DistinguishSpecialProbabilityDistributions
(This is the version I wrote for my other class: I may or may not have time to rewrite it a bit.)
Explanations of the names of the special distributions
Sampling distributions and the Central Limit Theorem (slideshow): Math2501-CentralLimitTheorem-slideshow
[I have more extensive notes on sampling distributions, statistics, and the three important theorems which I will post after a bit of a rewrite.]
Note: the standard deviation of the sample means is usually referred to as the standard error of the mean, to distinguish it from the standard deviation of the underlying random variable (the population standard deviation)
A statistic is a number that is computed from a random sample. Better: a statistic is a RV whose value depends on a random sample. So far we have two examples: the sample mean and the sample variance (and, by extension, the sample standard deviation is also a statistic). Generally, a statistic is connected to a population parameter which the statistic is intended to estimate: the sample mean $\bar{X}$ is connected to the population mean $\mu$, and the sample variance $S^{2}$ is connected to the population variance $\sigma^{2}$.
[If you are interested in the reason that the sample variance is divided by n-1 rather than n, here is a good discussion from stackexchange. The mathematical reason for it is that this makes the sample variance an “unbiased estimator” of the population variance. A complete mathematical explanation is here.]
Note: “Central” in the Central Limit Theorem (CLT) refers to the mean, which is a measure of “center” of a distribution. So this is a theorem about the limits of distributions of sample means.
We will not dwell on these three theorems in this section for very long. It is important to get the idea of what the CLT says and why it is important; try not to get overwhelmed by the formal notation. A good idea is to try to rephrase it in your own words.
An example I worked out to show what the Central Limit Theorem is saying about the distribution of sample means is in this spreadsheet. This is based on finite populations, but the results are still revealing. Some more examples from populations with very wild distributions are here. There is a demonstration program you can play with here. and some discussion and more examples here.
We will be using simulations in R to illustrate the CLT.
