Long page
descriptions

Chapter 2   Random Variables

2.1   Discrete random variables

2.1.1   Discrete distributions

A discrete random variable is usually one that can only take integer values — counts of something. Its distribution is fully described by the probabilities for these values, the variable's probability function.

2.1.2   Cumulative distribution function

The cumulative distribution function of a random variable, F(x), is the probability of getting a value less than or equal to x. For a discrete random variable, this is a step function.

2.1.3   Mean

A discrete random variable's mean is defined in a similar way to that of a discrete data set.

2.1.4   Expected values

The concept and definition of a variable's mean can be generalised to give the expected value of any function of its value.

2.1.5   Variance

A random variable's variance is the expected value of the squared distance to its mean. This summarises the spread of values in the distribution.

2.2   Continuous random variables

2.2.1   Probability density functions

A continuous random variable's distribution is described by a type of histogram with infinitely narrow classes, called the variable's probability density function.

2.2.2   Probability and area

The probability of a value within any range equals the probability density function's area above these values.

2.2.3   Mean and variance

The mean and variance of a continuous random variable are interpreted in a similar way to those of continuous variables, but are defined using integrals instead of summations.

2.2.4   Normal distributions

A particularly important type of continuous distribution is the family of normal distributions. These are symmetric distributions whose centre and spread are described by two parameters, μ and σ.

2.2.5   Important normal quantiles

A normal random variable has 90% probability of being within 1.645σ of μ, 95% probability of being within 1.96σ of μ, and 99% probability of being within 2.576σ of μ.

2.3   Discrete random samples

2.3.1   Independence

Two discrete random variables, X and Y, are independent if all events relating to the value of X are independent of events about Y.

2.3.2   Random samples

A random sample from a distribution is a collection of independent random variables, each of which has this distribution.

2.3.3   Linear functions of variables

Formulae are given for the mean and variance of a linear function of two independent random variables.

2.3.4   Properties of sums and means

The sum of values in a random sample is a random variable. Formulae for its mean and variance are given and similar formulae for the sample mean are also given.

2.3.5   Central limit theorem

The sum and mean of a random sample from a normal distribution both have normal distributions. For random samples from other distributions, the sum and mean have distributions that become close to normal as the sample size increases.

2.4   Uniform distribution

2.4.1   Family of uniform distributions

A random variable with equal probabilities for all integer values within some range, has a discrete uniform distribution.

2.4.2   Mean and variance

Formulae are given for the mean and variance of the discrete uniform distribution.

2.4.3   Sample mean

The distribution of the sample mean from a uniform distribution is found and shown to be close to a normal distribution when the sample size is large.