3.8  Bernoulli and Binomial Distributions

Until now, we have avoided mentioning any standard families of distributions such as the uniform, normal, or chi-squared families of distributions. This is intentional. All the results we have discussed so far are general and assume no particular distributions. We emphasized this fact by not incorporating any standard distributions into the discussion. Let’s reiterate. Formulas [3.30] and [3.31] for the mean and variance of a linear polynomial of a random vector are entirely general. So are principal component analysis, the fact that covariance matrices are positive semidefinite, and every other result we have presented so far. None assumes any particular distribution.

To construct probabilistic models, it is useful to consider standard families of distributions. In this and the next two sections, we discuss several families of distributions relevant for value-at-risk. We start with the Bernoulli and Binomial distributions. Primarily, we will use these in Chapter 12 when we discuss backtesting procedures. We have already used the Binomial distribution in our discussion of the Leavens PMMR in Section 1.7.1.

3.8.1 Bernoulli Distribution

A Bernoulli trial is an experiment with only two possible outcomes, which we may term “success” or “failure.” Tossing a coin is a Bernoulli trial: you can either get heads or tails. Playing the lottery is a Bernoulli trial: you will either win or lose. Inspecting a product on an assembly line to see if it is defective is a Bernoulli trial, as is applying for a job, proposing marriage or randomly selecting a ball from an urn containing blue and yellow balls. If just two candidates run for an elective office, the election is a Bernoulli trial.

Let p be the probability of “success,” in which case 1 – p is the probability of “failure.” Given a Bernoulli trial, define a random variable

[3.70]

This has a Bernoulli distribution, which has PF

[3.71]

The expectation, standard deviation, skewness, and kurtosis of a Bernoulli distribution are

[3.72]

[3.73]

[3.74]

[3.75]

3.8.2 Binomial Distribution

Perform m independent Bernoulli trials. The random variable X for the number of “successes” has a binomial distribution, which we denote B(m,p). The distribution has PF

[3.76]

where

[3.77]

and, for any non-negative integer n, n! is the factorial function:

[3.78]

The PF of a B(15, 0.6) distribution is shown in Exhibit 3.12.

Exhibit 3.12: A binomial distribution for 15 independent trials each having probability of “success” 0.6 is denoted B(15,0.6). Its probability function is shown above.

The expectation, standard deviation, skewness, and kurtosis of a B(m,p) distribution are

[3.79]

[3.80]

[3.81]

[3.82]

When m = 1, a binomial distribution reduces to a Bernoulli distribution. Using our existing notation, we denote a Bernoulli distribution B(1, p).