# APPENDIX A

# Technical Notes

## A.1 BINOMIAL DISTRIBUTION

A coin toss is an example of a *Bernoulli trial*, a random experiment with two possible outcomes. The coin is not necessarily stipulated to be “fair.” The probability of heads can be equal to any π ∊ [0,1]. If we assign the value *Y* = 1 to one of the outcomes of the Bernoulli trial and the value *Y* = 0 to the other, we say that *Y* follows a *Bernoulli distribution* with parameter π.

Suppose we repeat a Bernoulli trial *n* times and add up the resulting values of *Y*_{i} *i* = 1,…, *n*. Successive trials are independent. The random variable is said to follow a *binomial distribution* with parameters π and *n*. We have

The Bernoulli and binomial distributions are both discrete distributions. But the binomial distribution converges to the normal distribution as the number of trials *n* grows larger. This convergence result is an application of the *central limit theorem*. Specifically, if we standardize a binomially distributed random variable *X*, we can get its probability distribution arbitrarily close to that of a standard normal variate by increasing *n* enough:

This book uses the binomial distribution in two applications, to analyze the probability distribution ...

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