Signals and Systems/Common Distributions

There are a number of different random distributions in existence, many of which have been studied quite extensively, and many of which map very well to natural phenomena. This book will attempt to cover some of the most basic and most common distributions. This chapter will also introduce the idea of a distribution transformation, which can be used to turn a simple distribution into a more exotic distribution.

Uniform Distribution
One of the most simple distributions is a Uniform Distribution. Uniform Distributions are also very easy to model on a computer, and then they can be converted to other distribution types by a series of transforms.

A uniform distribution has a PDF that is a rectangle. This rectangle is centered about the mean, <&mu;x, has a width of A, and a height of 1/A. This definition ensures that the total area under the PDF is 1.

Gaussian Distribution
The Gaussian (or normal) distribution is simultaneously one of the most common distributions, and also one of the most difficult distributions to work with. The problem with the Gaussian distribution is that its pdf equation is non-integratable, and therefore there is no way to find a general equation for the cdf (although some approximations are available), and there is little or no way to directly calculate certain probabilities. However, there are ways to approximate these probabilities from the Gaussian pdf, and many of the common results have been tabulated in table-format. The function that finds the area under a part of the Gaussian curve (and therefore the probability of an event under that portion of the curve) is known as the Q function, and the results are tabulated in a Q table.

PDF and CDF
The PDF of a Gaussian random variable is defined as such:


 * $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$

The CDF of the Gaussian function is the integral of this, which any mathematician will tell you is impossible to express in terms of regular functions.

The Functions &Phi; and Q
The normal distribution with parameters &mu; = 0 and &sigma; = 1, the so-called standard normal distribution, plays an important role, because all other normal distributions may be derived from it. The CDF of the standard normal distribution is often indicated by &Phi;:


 * $$\Phi(x) = \frac{1}{2\pi}\int_{-\infty}^x e^{-\frac{t^2}{2}}dt$$.

It gives the probability for a standard normal distributed random variable to attain values less than x.

The Q function is the area under the right tail of the Gaussian curve and hence nothing more than 1 - &Phi;. The Q function is hence defined as:


 * $$Q(x) = 1 - \Phi(x)= \frac{1}{2\pi}\int_x^\infty e^{-\frac{t^2}{2}}dt$$

Mathematical texts might prefer to use the erf(x) and erfc(x) functions, which are similar. However this book (and engineering texts in general) will utilize the Q and Phi functions.

Poisson Distribution
The Poisson Distribution is different from the Gaussian and uniform distributions in that the Poisson Distribution only describes discrete data sets. For instance, if we wanted to model the number of telephone calls that are traveling through a given switch at one time, we cannot possibly count fractions of a phone call; phone calls come only in integer numbers. Also, you can't have a negative number of phone calls. It turns out that such situations can be easily modeled by a Poisson Distribution. Some general examples of Poisson Distribution random events are:


 * 1) The telephone calls arriving at a switch
 * 2) The internet data packets traveling through a given network
 * 3) The number of cars traveling through a given intersection

Transformations
If we have a random variable that follows a particular distribution, we would frequently like to transform that random process to use a different distribution. For instance, if we write a computer program that generates a uniform distribution of random numbers, and we would like to write one that generates a Gaussian distribution instead, we can feed the uniform numbers into a transform, and the output will be random numbers following a Gaussian distribution. Conversely, if we have a random variable in a strange, exotic distribution, and we would like to examine it using some of the easy, tabulated Gaussian distribution tools, we can transform it.