Probability, Random Variables, Statistics, and Random Processes

5Special Probability Distributions and Applications

There are infinitely many random variables, defined by their cdfs (equivalently pmfs or pdfs), out of which a few have real‐life applications and their own specific probability distributions. The focus of this chapter is on probability distributions that are important enough to have been given names, where there is a random experiment behind each of these distributions. After introducing some of the widely used discrete random variables, some of the well‐known continuous random variables are described. However, the Gaussian (normal) distribution, which is the most special random variable, gets its own chapter later. Few widely known applications are also highlighted in this chapter.

5.1 Special Discrete Random Variables

Discrete random variables are usually easy to understand intuitively, but sometimes unwieldy to analyze. There are dozens of discrete random variables, among which some well‐known and frequently encountered discrete probability distributions are now briefly discussed.

5.1.1 The Bernoulli Distribution

The Bernoulli distribution is a very simple, yet important, discrete random variable. The Bernoulli random variable X takes the value of 1 with probability p (also known as the probability of success) and the value of 0 with probability 1 − p (also known as the probability of failure), where 0 < p < 1. It is therefore a discrete random variable with the range {0, 1}. The pmf of a Bernoulli random variable is defined ...