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Chapter 7   Some Flexible Models

7.1   Over-dispersion of counts

7.1.1   Locations of items in space

Poisson processes can be used for events in time, and also for the location of items in 1- and 2-dimensional continua.

7.1.2   Overdispersion in Poisson distribution

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.

7.1.3   Generalised negative binomial distribution

A generalised negative binomial distribution can be used as a model for overdispersion of Poisson counts. Its mean and variance are given.

7.1.4   Overdispersion in binomial distribution

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.

7.1.5   Beta-binomial distribution

The beta-binomial distribution is presented as a generalisation of the binomial distribution that allows for overdispersion.

7.2   Varying hazard rate

7.2.1   Weibull distribution

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.

7.2.2   Mean, variance and shape

The Weibull distribution's mean and variance are derived.

7.2.3   Calculating Weibull probabilities

Two examples show how probabilities can be calculated for Weibull random variables.

7.3   Gamma distribution

7.3.1   Distribution for positive 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.

7.3.2   Gamma probabilities and quantiles

There are no formulae for cumulative probabilities or quantiles of Gamma distributions, but they can be easily evaluated with computer software such as Excel.

7.3.3   Some Gamma distribution properties

The Gamma distribution's mean and variance are derived. Adding two Gamma variables whose second parameters are equal also has a Gamma distribution.

7.4   Beta distribution

7.4.1   Values between zero and one

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.

7.4.2   Mean and variance

The mean and variance of the Beta distribution are derived.

7.5   Normal distribution

7.5.1   Standard normal distribution

The normal distribution's pdf is given and the standard normal distribution is defined.

7.5.2   Mean and variance

The mean and variance of the standard normal distribution are derived and used to find the mean and variance of other normal distributions.

7.5.3   Z-scores

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.

7.5.4   Probabilities for 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.

7.5.5   Normal quantiles

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.

7.5.6   Linear combinations, sums and means

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.

7.5.7   Independence of sample mean and variance

The mean and variance of a random sample from a normal distribution are independent.