Long page
descriptions

Chapter 3   Successes and Failures

3.1   Bernoulli distribution

3.1.1   Bernoulli trials

A Bernoulli random variable arises from a simple random experiment with two possible outcomes that are generically called success and failure. It has value 1 for a success and 0 for a failure.

3.1.2   Mean and variance

Formulae for the Bernoulli distribution's mean and variance are derived.

3.2   Binomial distribution

3.2.1   Binomial random variable

In a sequence of n independent Bernoulli trials that all have the same probability of success, the number of successes has a binomial distribution.

3.2.2   Binomial probability function

The probability function for the binomial distribution is derived.

3.2.3   Mean and variance

Formulae for the mean and variance of the binomial distribution are derived. The mean and variance of the proportion of successes can be found from them.

3.2.4   Shape of the binomial distribution

Bar charts show the shape of the binomial distribution for different values of n and π. The distribution's shape approaches a normal distribution when n increases.

3.2.5   Binomial examples

This page explains how to evaluate probabilities for binomial random variables and gives some practical examples.

3.2.6   Cumulative probabilities

The cumulative distribution function of the binomial distribution is graphed for different values of n and π.

3.3   Geometric distribution

3.3.1   Geometric random variable

The number of Bernoulli trials until the first success has a geometric distribution. Its probabilitiy function is found.

3.3.2   Cumulative distribution function

The probability of a value less than or equal to x has a particularly simple formula.

3.3.3   Examples

An example is described that relates to the number of times a six-sided die must be rolled until the first six is observed.

3.3.4   Mean and variance

Formulae for the mean and variance of the geometric distribution are derived.

3.4   Negative binomial distribution

3.4.1   Probability function

The number of independent success/failure trials until the k'th success has a negative binomial distribution. This is a generalisation of the geometric distribution.

3.4.2   Alternative definition

The number of failures until the k'th success equals the number of trials, minus k. It is also sometimes said to have a negative binomial distribution.

3.4.3   Finding negative binomial probabilities

An example is given to show how negative binomial probabilities can be calculated.

3.4.4   Cumulative distribution function

Cumulative probabilities can either be evaluated by summing probabilities from this distribution, or by summing binomial probabilities, which is usually a simpler calculation.

3.4.5   Mean and variance

The mean and variance of the negative binomial distribution are derived.