O'Reilly logo

Financial Risk Management: Models, History, and Institutions by Allan M. Malz

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 Yi i = 1,…, n. Successive trials are independent. The random variable images is said to follow a binomial distribution with parameters π and n. We have

images

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:

images

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required