What is a random variable in statistics?

Definition

A random variable is a function that assigns a numerical value to each outcome in a sample space of a random experiment. It provides a way to quantify uncertain outcomes. There are two main types:

• Discrete random variable: Takes on a countable set of values (e.g., number of heads in coin tosses).
• Continuous random variable: Takes on values from an interval or collection of intervals (e.g., height, weight).

Mathematically, if $S$ is the sample space, a random variable $X$ is a function $X: S \to mathbb{R}$.

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Worked Example

Example: Suppose you toss a fair coin twice. Let $X$ be the random variable representing the number of heads observed.

Step 1: List the sample space

The possible outcomes are:

• HH
• HT
• TH
• TT

Step 2: Assign values of $X$

• $X(\text{HH}) = 2$
• $X(\text{HT}) = 1$
• $X(\text{TH}) = 1$
• $X(\text{TT}) = 0$

Step 3: Find the probability distribution of $X$

• $P(X = 0) = P(\text{TT}) = \frac{1}{4}$
• $P(X = 1) = P(\text{HT}) + P(\text{TH}) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$
• $P(X = 2) = P(\text{HH}) = \frac{1}{4}$

So, the probability distribution is:

\[\begin{array}{c|c} x & P(X = x) \\ hline 0 & \frac{1}{4} \\ 1 & \frac{1}{2} \\ 2 & \frac{1}{4} \\ \end{array}\]

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Takeaways

• A random variable maps outcomes of a random process to numbers.
• It can be discrete (countable values) or continuous (any value in an interval).
• Random variables allow us to analyze and compute probabilities for numerical outcomes.