Some Important Continuous Distributions

This chapter provides examples that present the important properties of the uniform distribution. The first of these shows that a linear transformation of an uniform random variable is again uniform; the second relates the distribution function of the uniform distribution over the unit interval to the distribution function of an arbitrary continuous random variable. Among all probability distributions, the normal distribution has a paramount role and is the most commonly used distribution in a wide array of applications. The chapter establishes the theoretical framework of random variables and presents its main properties, some of which are unique among distributions on the real line. It argues that the normal distribution may be used as an approximation to the Poisson distribution. Calculation of probabilities associated with the standard normal distribution requires evaluation of the distribution function. An important property of the exponential distribution is the lack of memory property of the distribution.