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.
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.
A discrete random variable's mean is defined in a similar way to that of a discrete data set.
The concept and definition of a variable's mean can be generalised to give the expected value of any function of its value.
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.
A continuous random variable's distribution is described by a type of histogram with infinitely narrow classes, called the variable's probability density function.
The probability of a value within any range equals the probability density function's area above these values.
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.
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.
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.
Two discrete random variables, X and Y, are independent if all events relating to the value of X are independent of events about Y.
A random sample from a distribution is a collection of independent random variables, each of which has this distribution.
Formulae are given for the mean and variance of a linear function of two independent random variables.
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.
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.
A random variable with equal probabilities for all integer values within some range, has a discrete uniform distribution.
Formulae are given for the mean and variance of the discrete uniform distribution.
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.