Random Variables | Introduction to Probability

What is a Random Variable?

A random variable $X$ is a function from the sample space to the real numbers:

$X: \Omega \to \mathbb{R}$

Formally, it assigns a numerical value to each outcome.

Why Functions?

Outcomes (like “three coin flips”) are often complex or qualitative. Random variables let us work numerically: counting heads, summing dice, measuring time.

Example: Coin Flips

Flip three coins. The outcome is a sequence like $\{H, T, H\}$ . Define $X$ = number of heads.

Outcome	$X$
HHH	3
HHT	2
HTH	2
HTT	1
THH	2
THT	1
TTH	1
TTT	0

Discrete vs Continuous Random Variables

Discrete

$X$ is discrete if it takes values from a finite or countably infinite set.

Examples:

Number of heads in 10 flips: $\{0, 1, 2, \ldots, 10\}$
Rolls until first 6: $\{1, 2, 3, \ldots\}$
Number of customers arriving: $\{0, 1, 2, \ldots\}$

Continuous

$X$ is continuous if it can take any value in an interval.

Examples:

Time until failure: $[0, \infty)$
Temperature: $\mathbb{R}$
Height: $[0, 300]$ cm

Indicator Variables

An indicator variable (or Bernoulli variable) for event $E$ is:

$I_E = \begin{cases} 1 & \text{if } E \text{ occurs} \\ 0 & \text{otherwise} \end{cases}$

Properties

$E[I_E] = 0 \cdot P(E^c) + 1 \cdot P(E) = P(E)$

The expected value of an indicator equals the probability of the event.

This is incredibly useful: to compute $P(E)$ , compute $E[I_E]$ .

Example: Matching Cards

Shuffle a deck. What is the expected number of cards in their original position?

Let $I_i$ indicate card $i$ is in position $i$ . Then $X = \sum_{i=1}^{52} I_i$ counts matches.

$E[I_i] = P(\text{card } i \text{ in position } i) = \frac{1}{52}$

By linearity:

$E[X] = \sum_{i=1}^{52} E[I_i] = 52 \times \frac{1}{52} = 1$

On average, exactly one card is in its original position, regardless of deck size.

Types of Random Variables

Type	Values	Example
Indicator	$\{0, 1\}$	Success/failure
Binary	$\{0, 1\}$ (or $\{-1, 1\}$ )	Yes/no outcomes
Count	$\{0, 1, 2, \ldots\}$	Number of arrivals
Continuous	Real interval	Time, weight

Random Variables in Programming

Think of a random variable like a variable in a programming language:

It has a type (discrete or continuous)
It has a range of possible values
It has a distribution describing likelihood of each value
It can be sampled to produce a concrete value

Example in Context

# X is a random variable
# - Type: discrete
# - Range: {1, 2, 3, 4, 5, 6}
# - Distribution: uniform
import random
x = random.randint(1, 6)  # Sample a value

Functions of Random Variables

If $X$ is a random variable and $g$ is a function, then $Y = g(X)$ is also a random variable.

Example: If $X$ is the roll of a die, then $Y = 2X + 1$ takes values $\{3, 5, 7, 9, 11, 13\}$ .

Summary

Concept	Definition
Random variable	Function $\Omega \to \mathbb{R}$
Discrete RV	Takes countably many values
Continuous RV	Takes values in an interval
Indicator	$I_E = \begin{cases} 1 & E \text{ occurs} \\ 0 & \text{otherwise} \end{cases}$