Discrete Random Variables

Summary :

A discrete random variable takes countable, separate values, each with a probability. This chapter defines the probability distribution function for a discrete variable, shows how to find its expected value and standard deviation, and introduces the main discrete models: the binomial, geometric, hypergeometric, and Poisson distributions.

The Probability Distribution Function

A discrete random variable takes values that can be counted, such as the number of heads in ten coin tosses. Its probability distribution function lists each possible value together with the probability of that value occurring. Two conditions must hold: every probability lies between zero and one, and the probabilities of all outcomes add up to one. Presented as a table, the distribution gives a complete description of how likely each outcome is.

Expected Value and Standard Deviation

The expected value of a discrete random variable is its long-run average, found by multiplying each value by its probability and adding the results. It is the mean of the distribution and need not equal any single possible outcome. The standard deviation measures how much the values typically differ from the expected value, calculated from the squared distances of each value from the mean weighted by their probabilities. Together they summarise the centre and spread of the distribution.

Common Discrete Distributions

Several named distributions describe common situations. The binomial distribution counts successes in a fixed number of independent trials, each with the same probability of success. The geometric distribution counts the trials needed to get the first success. The hypergeometric distribution applies when sampling without replacement from two groups, and the Poisson distribution models the number of events in a fixed interval when they occur independently at a constant average rate.