A1.4 Continuous Probability Distributions

A1.4 Continuous Probability Distributions

A1.4.1 The Normal Distribution

A teacher of mine once observed that there never is, nor ever was, such a naturally occurring thing as a normal probability density function. However, this distribution is the foundation of much of the discipline of statistics. If there were such an artifact in Nature, it would have a pdf like this:

The mean of this distribution is μ and its variance is σ. This is an interesting distribution in that it is completely specified by just these two parameters. While we do know that:

there are not too many random variables in Nature that have this range (-∞, ∞). For example, it is often stated that the height of the population of the country is normally distributed. This being the case, we would certainly expect to find some individuals with a height of -200 cm or perhaps even 2000 cm. This, however, is not the case. People do not come in those heights.

In that a truly normal distribution is an improbable event in its own right, we must learn to adjust to this fact. The assumption of normality is very important. What we must learn to think about is not that the population we are modeling is not really normal. Rather, we must always worry about how robust the statistical procedures we are using are in relation to potential departures from the assumption of normality. Some techniques are very sensitive about these departures. Other techniques are not so sensitive and, as a result, are said to be robust.

A1.4.2 The χ² Distribution

The χ² distribution scarcely has a life of its own. Only rarely will this distribution be used by itself for modeling purposes. We will use it primarily as a building block for the t distribution and the F distribution, both of which we will use frequently. The pdf

is a χ² distribution with k degrees of freedom. This distribution is defined on the interval [0, ∞).

The mean of this distribution and its variance depend entirely on the degrees of freedom of the distribution as follows:

μ = k

σ² = 2k

There is an interesting relationship between the χ² distribution and the normal distribution. That is, if the random variable X has a normal distribution with a mean of μ and a variance is σ, then the random variable

has a χ² distribution with one degree of freedom.

A1.4.3 The t Distribution

Let W be a random variable where W ~ N(0,1) and V be a random variable that has a χ² distribution with k degrees of freedom. Under the condition that W and V are stochastically independent, we can define a new random variable T where

A t distribution with n degrees of freedom can be defined as that of the random variable T symmetrically distributed about 0. Then, f(x) = Pr(T = x) and

A1.4.4 The F Distribution

Now consider two independently distributed random variables U and V having n₁ and n₂ degrees of freedom, respectively. We can define a new random variable

Then,

f(x) = Pr(F = x)

and

A1.4.5 The Beta Distribution

A random variable X has a beta distribution with parameters α and β (α, β > 0) if X has a continuous distribution whose pdf is:

The expected value of X is given by:

and the variance of X is given by:

This distribution will be particularly useful because of its relationship to the binomial distribution, as shown in Chapter 10.

A1.4.6 The Dirichlet Distribution

The pdf for a Dirichlet distribution, D(α, α_T), with a parametric vector α = (α₁,α₂,...,α_M), where (α_i > 0; i = 1,2,...,M), is:

where (w_i > 0; i = 1,2,...,M) and . The expected values of the w_i are given by:

where . In this context, α₀ represents the total epochs. The variance of the w_i is given by:

and the covariance is given by:

This distribution will be particularly useful because of its relationship to the multinomial distribution as was shown in Chapter 10.