Some Special Distributions

Uniform Distribution
The continuous uniform distribution on an interval of R is one of the simplest of all probability distributions, but nonetheless very important. In particular, continuous uniform distributions are the basic tools for simulating other probability distributions. The uniform distribution corresponds to picking a point at random from the interval. The uniform distribution on an interval is a special case of the general uniform distribution with respect to a measure, in this case Lebesgue measure (length) on R.
The Standard Uniform Distribution
The uniform distribution on the interval [0,1] is known as the standard uniform distribution. A simulation of a random variable with the standard uniform distribution is known in computer science as a random number. All programming languages have functions for computing random numbers, as do calculators, spreadsheets, and mathematical and statistical software packages.
Distribution Functions
By definition, the uniform distribution on an interval has constant density on that interval. Hence random variable U has the standard uniform distribution if U has probability density function g(u)=1 for 0?u?1. Since the density function is constant, the mode is not meaningful.
The Gamma Distribution
In this section we will study a family of distributions that has special importance in probability statistics. In particular, the arrival times in the Poisson process have gamma distributions, and the chi-square distribution is a special case of the gamma distribution.
The Gamma Function
The gamma function, first introduced by Leonhard Euler, is defined as follows
?(k)=? sk?1e?sds,k?(0,?)
The gamma function is well defined, that is, the integral in the gamma function converges for any k>0.
The Chi-Square Distribution
In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of the sample variance when the underlying distribution is normal and in goodness of fit tests.
The Density Function
For n>0, the gamma distribution with shape parameter k=n2 and scale parameter 2 is called the chi-square distribution with n degrees of freedom. For reasons that will be clear later, n is usually a positive integer, although technically this is not a mathematical requirement.
The chi-square distribution with n degrees of freedom has probability density function
f(x)={1/[2n/2?(n/2)]} xn/2-1e?x/2,0
We say a random variable X has a normal distribution if its pdf is given as below. The parameters ? and ?2 are the mean and variance of X respectively. We write X has N(?,?2).