Class summary: Friday, 4/11

1. Normal approximation to the binomial distribution versus normal approximation to sums of i.i.d. r.v.'s. These involve two distinct formulas, and it is important to known which one to apply in a given situation. The former (approximation to the binomial distribution) came up much earlier in this class in connection with repeated independent S/F trials. It applies only in situations that can be modeled as independent S/F trials. By contrast, the latter formula applies in much more general contexts, namely, whenever a sum of (a large number of) i.i.d. random variables is involved.
2. The law of large numbers (or law of averages). This "law", which is a consequence of the normal approximation formula, states that the averages (in the naive sense) of a large number of i.i.d. random variables tend to "hone in" on the expectation of these r.v.'s. It can be used to determine the expectation of an unknown r.v. (e.g., a die with unknown bias) experimentally, by repeating the experiment a large number of times and taking the average of the observed values (e.g., the numbers on the die showing up).
3. The square root law. This is a rule of thumb saying that the sums of n i.i.d. random variables have a standard deviation proportional to the square root of n, and the averages of n i.i.d. random variables have a standard deviation inverse proportional to the square root of n. The standard deviation is a measure of the "spread" of a distribution, or the "statistical noise" associated with a distribution. The square root law can be used to make rough estimates and, for instance, or dismiss certain hypotheses. For example, in a million coin tosses the number of heads and tails can be expected to differ by a few thousand due to "statistical noise". However, a difference of a few hundred thousand (or even ten thousand) is extremely unlikely if the coin is fair, so in that case one would have to conclude that the coin is almost certainly biased.

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