Discrete Random Variables

What is a Random Variable?

A random variable is a function that assigns a numerical value to each outcome in the sample space.

Definition

Mathematically, a random variable $X$ is a function from the sample space $\Omega$ to the real numbers:

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

For each outcome $\omega \in \Omega$, the random variable $X$ assigns a numerical value $X(\omega)$.

Example: Coin Flips

Flip a coin 3 times. Sample space: $\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$

Let $X$ = number of heads. Then:

• $X(HHH) = 3$
• $X(HHT) = 2$
• $X(HTT) = 1$
• $X(TTT) = 0$

Discrete vs Continuous

A random variable is discrete if it takes values in a countable set:

• Finite: $\{1, 2, 3, 4, 5, 6\}$ (die roll)
• Infinite but countable: $\{1, 2, 3, \ldots\}$ (number of coin flips until heads)

A random variable is continuous if it takes values in an interval:

• $[0, 1]$ (uniform on unit interval)
• $(-\infty, \infty)$ (normal distribution)

Probability Mass Function (PMF)

The PMF gives the probability of each possible value:

$$p_X(x) = P(X = x)$$

Properties

1. $p_X(x) \geq 0$ for all $x$
2. $\sum_x p_X(x) = 1$ (total probability is 1)

Example: Fair Die

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

Example: Bernoulli

$X$ takes values 0 or 1 with probabilities $1-p$ and $p$:

$$p_X(0) = 1-p, \quad p_X(1) = p$$

Common Discrete Distributions

Bernoulli

Single trial with success probability $p$:

• $P(X = 1) = p$
• $P(X = 0) = 1 - p$

Binomial

$n$ independent Bernoulli trials:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Geometric

Number of trials until first success:

$$P(X = k) = (1-p)^{k-1} p \quad \text{for } k = 1, 2, 3, \ldots$$

Poisson

Number of events in a fixed interval:

$$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

Calculating Probabilities

For any event $A$:

$$P(X \in A) = \sum_{x \in A} p_X(x)$$

Example

For a fair die, find $P(X \geq 4)$:

$$P(X \geq 4) = p_X(4) + p_X(5) + p_X(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}$$