Random Variable

In many situations arising in business and economics, the outcome of a random experiment is more useful to us as a number than as a description in words. For example, when three items are picked at random for quality inspection, we are usually interested in how many of them are defective, not in which particular items were picked. A random variable is exactly this kind of number — a rule that assigns a real number to every outcome of a random experiment.

Formally, if $S$ is the sample space of a random experiment, a random variable $X$ is a function $X : S \to \mathbb{R}$ that assigns a real number $X(s)$ to every outcome $s \in S$. By convention we denote the random variable itself by a capital letter ($X$, $Y$, ...) and a particular value it can take by the corresponding small letter ($x$, $y$, ...). The event "$X$ takes the value $x$" is written $X = x$, and its probability is written $P(X = x)$.

Discrete random variable. A random variable that can take only a finite number of values, or a countably infinite list of values (like $0, 1, 2, 3, \ldots$), is called a discrete random variable. Examples: the number of heads when three coins are tossed ($X = 0,1,2,3$); the number of defective bulbs in a box of 10; the number of customers arriving at a shop counter in an hour.

Continuous random variable. A random variable that can take any value in an interval of real numbers — such as the height of a student, the weight of a packet, or the time taken to serve a customer — is called a continuous random variable. This chapter deals only with discrete random variables and the probability distributions built on them; continuous distributions (like the Normal distribution) are studied separately.

This part of the Gujarat Std-12 Statistics (Business Mathematics & Statistics) syllabus builds directly on the ideas of probability studied earlier — a random variable is simply a convenient numerical label for the outcomes whose probabilities we already know how to compute.