Poisson processes can be used for events in time, and also for the location of items in 1- and 2-dimensional continua.
The assumptions underlying a Poisson distribution are not always satisfied. If the rate of events, λ, varies, the distribution of event counts has a larger variance than would arise in a Poisson distribution.
A generalised negative binomial distribution can be used as a model for overdispersion of Poisson counts. Its mean and variance are given.
The binomial distribution assumes that all success/failure trials are independent and have the same probability of success. If these assumptions are violated, the distribution has a larger variance than a binomial distribution.
The beta-binomial distribution is presented as a generalisation of the binomial distribution that allows for overdispersion.
An item's lifetime has an exponential distribution if its hazard rate remains constant. The Weibull distribution allows the hazard rate to increase or decrease over time.
The Weibull distribution's mean and variance are derived.
Two examples show how probabilities can be calculated for Weibull random variables.
The Gamma distribution is another two-parameter distribution that can be used for variables whose values cannot be negative. Its parameters give it similar flexibility in shape to the Weibull distribution but its upper tail is longer.
There are no formulae for cumulative probabilities or quantiles of Gamma distributions, but they can be easily evaluated with computer software such as Excel.
The Gamma distribution's mean and variance are derived. Adding two Gamma variables whose second parameters are equal also has a Gamma distribution.
Sometimes the value of a random variable must lie between zero and one. The family of Beta distributions is flexible enough to model many such variables.
The mean and variance of the Beta distribution are derived.
The normal distribution's pdf is given and the standard normal distribution is defined.
The mean and variance of the standard normal distribution are derived and used to find the mean and variance of other normal distributions.
All normal distributions have the same shape when expressed in terms of standard deviations from the mean; these are called z-scores and have standard normal distributions.
Excel can be used to find probabilities about normal distributions. Translating questions into ones about z-scores gives a method to find probabilities using tables of probabilities for the standard normal distribution.
We usually want to find a cumulative probability corresponding to a given x-value. Occasionally the x-value corresponding to a given cumulative probability is needed — a quantile of the distribution.
A linear combination of two independent normal variables also has a normal distribution. So do the mean and sum of the values in a random sample.
The mean and variance of a random sample from a normal distribution are independent.