Random Variables

Overview

A random variable is a function that assigns a numerical value to each outcome in a sample space. It bridges probability theory with real-world data.

Types of Random Variables

Type	Description	Examples
Discrete	Countable values	Dice roll, number of defects
Continuous	Any value in an interval	Height, weight, time

Discrete Random Variables

Probability Mass Function (PMF)

For discrete random variable $X$:

$$P(X = x) = f(x)$$

Properties:

• $f(x) \geq 0$ for all $x$
• $\sum f(x) = 1$

Cumulative Distribution Function (CDF)

$$F(x) = P(X \leq x) = \sum_{t \leq x} f(t)$$

Properties:

• $0 \leq F(x) \leq 1$
• $F(x)$ is non-decreasing
• $F(-\infty) = 0$, $F(\infty) = 1$

Continuous Random Variables

Probability Density Function (PDF)

$$P(a \leq X \leq b) = \int_a^b f(x) \, dx$$

Properties:

• $f(x) \geq 0$ for all $x$
• $\int_{-\infty}^{\infty} f(x) \, dx = 1$
• $P(X = x) = 0$ for any single point

CDF for Continuous Variables

$$F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt$$

Relationship:

$$f(x) = \frac{dF(x)}{dx}$$

Key Formulas Summary

Concept	Discrete	Continuous
Probability	$P(X = x) = f(x)$	$P(a < X < b) = \int f(x)dx$
CDF	$F(x) = \sum f(t)$	$F(x) = \int f(t)dt$
Sum/Integral	$\sum f(x) = 1$	$\int f(x)dx = 1$

Examples

Example 1: Discrete PMF

Rolling a fair die. $X$ = number on die.

$$P(X = x) = \frac{1}{6} \quad \text{for } x = 1, 2, 3, 4, 5, 6$$

Example 2: Discrete CDF

For the die:

$$F(3) = P(X \leq 3) = P(1) + P(2) + P(3) = \frac{3}{6} = 0.5$$

Example 3: Continuous PDF

If $f(x) = 2x$ for $0 \leq x \leq 1$:

$$P(0 \leq X \leq 0.5) = \int_0^{0.5} 2x \, dx = \left[x^2\right]_0^{0.5} = 0.25$$

Example 4: Creating a PMF Table

Two coins flipped. $X$ = number of heads.

$x$	0	1	2
$P(X=x)$	0.25	0.50	0.25

Joint Distributions

For two random variables $X$ and $Y$:

Discrete

$$P(X = x, Y = y) = f(x, y)$$

Marginal Distributions

$$P(X = x) = \sum_y P(X = x, Y = y)$$

Independence

$X$ and $Y$ are independent if:

$$P(X = x, Y = y) = P(X = x) \times P(Y = y)$$

Common Notation

Symbol	Meaning
$X, Y$	Random variables
$x, y$	Specific values
$f(x)$	PMF or PDF
$F(x)$	CDF
$E(X)$	Expected value
$\text{Var}(X)$	Variance