Uniform distribution discrete probability distributions, random

For discrete random variables, the probability distribution can often be determined using mathematical intuition, as all experiments are characterized by a hxed set of outcomes. For example, consider an experiment in which a six-sided die is thrown. The variable x denotes the number on the die face, and P(x) is the probability distribution, i.e., the chance of observing x on the face following a throw. Since this random variable is discrete, if the die is fair, then the probability of any possible value is equal and is given by 1/n, where n is the number of sides, since the sum of all possible outcomes must be unity. The distribution is, therefore, called uniform. Hence, for n = 6, P(x) = 1/6, where x = 1, 2, 3, 4, 5, or 6. 

Uniform Distribution n A probability distribution where the probability in the case of a discrete random variable or the density function in the case of a continuous random variable, X, with values, x, are constant (equal) over an interval, a,b), where x is greater than or equal to a and X less than or equal to b and x is zero outside the interval. The uniform distribution is sometimes referred to as the rectangular distribution since, a plot of its probability or density function resembles a rectangle. When the random variable, X, is discrete, the uniform distribution is referred to as the discrete uniform distribution and has the probability, P(x. ), with the form of ... 

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fiirth), we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. 

Namdev et al. (1991) simulated the fluctoating dissolved oxygen concentrations in large bioreactors in a 2 L vessel using a Monte Carlo approach. A lognormal distribution, described by a mean circulation of 20 s and a standard deviation of 8.9 s, was discretized into n elements of equal probability, each with a corresponding circulation time. A uniform random number was then used to select a circulation time. Therefore, a random circulation time was selected... 

---