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Chapter 10   Normal-based distributions

10.1   Chi-squared distribution

10.1.1   Chi-squared with 1 degree of freedom

The square of a variable with a standard normal distribution has a chi-squared distribution with one degree of freedom. It is a special case of a gamma distribution.

10.1.2   Properties of Chi-squared (1 df)

This chi-squared distribution is extremely skew. Its mean and variance are found from those of the gamma distribution.

10.1.3   General chi-squared distribution

The sum of n independent chi-squared variables (1 df) has a chi-squared (n df) distribution. It is also a special case of a gamma distribution.

10.1.4   General chi-squared properties

The general chi-squared distribution with n degrees of freedom is also skew but becomes closer to symmetric when n increases. Its mean and variance are also given.

10.1.5   Sample variance

The variance of a random sample from a normal distribution has a distribution that is proportional to a chi-squared distribution.

10.1.6   Confidence interval for normal variance

A pivot can be based on a normal sample's variance. A confidence interval for the underlying normal distribution's variance σ² can be found from it.

10.1.7   Pooled variance from several samples

The sample variances from several normal random samples can be combined. If the underlying normal variances are equal, this pooled variance has a chi-squared distribution, allowing a confidence interval for σ² to be found.

10.2   T distribution

10.2.1   Definition of t distribution

The ratio of a standard normal variable and the square root of an independent chi-squared variable (divided by its degrees of freedom) has a t distribution. Its pdf is given.

10.2.2   Mean, variance and shape

Formulae for the mean and variance of the t distribution are given. Its tails are longer than those of a standard normal distribution, but the two distributions' shapes are close when the degrees of freedom are large.

10.2.3   Pivot for normal mean

A function of the sample mean, sample variance and μ is shown to have a t distribution if the sample comes from a normal distribution. It is a pivot for μ.

10.2.4   Confidence interval for normal mean

The pivot on the previous page can be used to find a confidence interval for a normal distribution's mean, μ.

10.2.5   Two samples from normal distributions

The difference between two sample means from normal distributions is normally distributed. The "pooled" estimator of the common variance has a chi-squared distribution.

10.2.6   Estimating difference between means

A pivot can be found from the difference between the sample means and pooled estimate of the variance. This can be used to find a confidence interval for the difference between the means.

10.3   F distribution

10.3.1   Ratio of chi-squared variables

The ratio of two independent chi-squared variables (divided by their degrees of freedom) has an F distribution. Its probability density function is shown.

10.3.2   Mean, variance and shape

Formulae are given for the F distribution's mean and variance. It is very skew unless both of its degrees of freedom are large.

10.3.3   Comparing sample variances

The ratio of two sample variances from normal distributions has a distribution proportional to an F distribution. A confidence interval for the ratio of the normal variances can be found from it.

10.3.4   Relationship between t and F distributions

The square of a random variable with a t distribution has an F distribution.