Financial Risk Management: Models, History, and Institutions

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 ...