PSTAT 5A: Discrete Random Variables

Lecture 8

Narjes Mathlouthi

2025-07-10

Welcome to Lecture 8

Discrete Random Variables

From outcomes to numbers: quantifying randomness

Today’s Learning Objectives

By the end of this lecture, you will be able to:

  • Define random variables and distinguish discrete from continuous
  • Work with probability mass functions (PMFs) and cumulative distribution functions (CDFs)
  • Calculate expected values and variances for discrete random variables
  • Apply properties of expectation and variance
  • Work with common discrete distributions (Bernoulli, Binomial, Geometric, Poisson)
  • Solve real-world problems using discrete random variables
  • Use python to compute probabilities and parameters

What is a Random Variable?

Definition: A random variable is a function that assigns a numerical value to each outcome of a random experiment.

Notation: Usually denoted by capital letters \(X\), \(Y\), \(Z\)

Key insight: Random variables transform outcomes into numbers, making statistical analysis possible

Treena - Random Variables

Die Roll Example: Mapping Outcomes to Numbers

1 2 3 4 5 6

Random Variable X maps each die face to its numerical value.

Why Use Random Variables?

Random variables allow us to:

  • Quantify outcomes numerically
  • Calculate means, variances, and other statistics
  • Model real-world phenomena
  • Make predictions and decisions
  • Compare different random processes

Examples: Height, test scores, number of defects, wait times, stock prices

Treena - Random Variables

Types of Random Variables

Today we focus on discrete random variables - notice there are gaps between possible values!


Discrete vs. Continuous: Demystifying the type of Random Variables

Discrete Random Variable

Takes on a countable number of values

Can list all possible values

Examples:
• Dice rolls: {1, 2, 3, 4, 5, 6}
• Number of emails: {0, 1, 2, 3, …}
• Quiz scores: {0, 1, 2, …, 10}

Continuous Random Variable

Takes on uncountably many values

Cannot list all possible values

Examples:
• Height: Any value in \([0, \infty)\)
• Time: Any positive real number
• Temperature: Any real number

:::

Probability Mass Function (PMF)

Definition: The Probability Mass Function (PMF) of a discrete random variable \(X\) is:

\[P(X = x) = \text{probability that } X \text{ takes the value } x\]

Properties of PMF:

  • \(P(X = x) \geq 0\) for all \(x\)
  • \(\sum_{\text{all } x} P(X = x) = 1\)

Interactive PMF Demo: Fair Die

Theoretical vs Observed Frequencies

Statistics: Click “Roll Die” to see statistics

PMF Example: Two Coin Flips

Two Coin Flips - Number of Heads

Let \(X\) = number of heads in two coin flips

Sample Space: \(\{HH, HT, TH, TT\}\)

H T H T

Current outcome: H T H T

Click coins to flip them!

Table summarizes by number of heads, not the exact sequence.
\(x\) (heads) Outcomes \(P(X = x)\) Empirical
0 TT 0.25 0
1 HT, TH 0.50 0
2 HH 0.25 0
Empirical probability updates as you flip!

Cumulative Distribution Function (CDF)

The cumulative distribution function of a random variable \(X\) is:

\[F(x) = P(X \leq x)\]

Properties of CDF: 1. \(F(x)\) is non-decreasing 2. \(\lim_{x \to -\infty} F(x) = 0\) 3. \(\lim_{x \to \infty} F(x) = 1\) 4. \(F(x)\) is right-continuous

Expected Value and Variance

The expected value of a discrete random variable \(X\) is:

\[E[X] = \mu = \sum_{\text{all } x} x \cdot P(X = x)\]

The variance of a random variable \(X\) measures spread around the mean:

\[\text{Var}(X) = \sigma^2 = E[(X - \mu)^2] = E[X^2] - (E[X])^2\]

Expected value represents the long-run average if we repeat the experiment many times.

Law of Large Numbers Demo

Law of Large Numbers Demo

Watch how the sample mean converges to the expected value! {.smaller}

Common Discrete Distributions

Bernoulli

Single trial, two outcomes

Parameters: \(p\) (success probability)

PMF: \(P(X = 1) = p\), \(P(X = 0) = 1-p\)

Mean: \(p\)

Variance: \(p(1-p)\)

Binomial

\(n\) independent Bernoulli trials

Parameters: \(n\) (trials), \(p\) (success prob.)

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

Mean: \(np\)

Variance: \(np(1-p)\)

Geometric

Trials until first success

Parameters: \(p\) (success probability)

PMF: \(P(X = k) = (1-p)^{k-1} p\)

Mean: \(1/p\)

Variance: \((1-p)/p^2\)

Poisson

Events in fixed interval

Parameters: \(\lambda\) (average rate)

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

Mean: \(\lambda\)

Variance: \(\lambda\)

:::

Interactive Distribution Explorer

Distribution Visualizer

Practice Problem 1

A box contains 3 red balls and 2 blue balls. Two balls are drawn without replacement. Let \(X\) = number of red balls drawn. Find the PMF of \(X\).

Solution. \(X\) can take values 0, 1, or 2.

\(P(X = 0) = \frac{\binom{3}{0}\binom{2}{2}}{\binom{5}{2}} = \frac{1 \times 1}{10} = \frac{1}{10}\)

\(P(X = 1) = \frac{\binom{3}{1}\binom{2}{1}}{\binom{5}{2}} = \frac{3 \times 2}{10} = \frac{6}{10}\)

\(P(X = 2) = \frac{\binom{3}{2}\binom{2}{0}}{\binom{5}{2}} = \frac{3 \times 1}{10} = \frac{3}{10}\)

Check: \(\frac{1}{10} + \frac{6}{10} + \frac{3}{10} = 1\) ✓

Practice Problem 2: Expected Value

Using the red balls example from Problem 1, find \(E[X]\) and \(\text{Var}(X)\).

Solution. Expected Value: \[E[X] = 0 \times \frac{1}{10} + 1 \times \frac{6}{10} + 2 \times \frac{3}{10} = 0 + \frac{6}{10} + \frac{6}{10} = 1.2\]

Variance: \[E[X^2] = 0^2 \times \frac{1}{10} + 1^2 \times \frac{6}{10} + 2^2 \times \frac{3}{10} = 0 + \frac{6}{10} + \frac{12}{10} = 1.8\]

\[\text{Var}(X) = E[X^2] - (E[X])^2 = 1.8 - (1.2)^2 = 1.8 - 1.44 = 0.36\]

Standard Deviation: \(\sigma = \sqrt{0.36} = 0.6\)

Practice Problem 3

A student takes a 10-question multiple choice quiz with 4 options per question. If the student guesses randomly, what’s the probability of getting exactly 3 correct?

Solution. This is a binomial distribution with \(n = 10\), \(p = 1/4 = 0.25\)

\[P(X = 3) = \binom{10}{3} \times (0.25)^3 \times (0.75)^7\]

\[P(X = 3) = 120 \times 0.015625 \times 0.1335 \approx 0.2503\]

So there’s about a 25% chance of getting exactly 3 correct by guessing.

Properties of Expected Value

Linearity of Expectation

  1. \(E[c] = c\) (constant)
  2. \(E[cX] = c \cdot E[X]\) (scaling)
  3. \(E[X + Y] = E[X] + E[Y]\) (additivity)
  4. \(E[aX + bY + c] = aE[X] + bE[Y] + c\)

Variance Properties

  • \(\text{Var}(aX + b) = a^2 \text{Var}(X)\)
  • \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)\) (if \(X\) and \(Y\) are independent)

Important: Property 3 holds even if \(X\) and \(Y\) are dependent!

Key Takeaways

Main Concepts

  • Random variables transform outcomes into numbers for mathematical analysis
  • PMF gives probabilities for specific values; CDF gives cumulative probabilities
  • Expected value is the long-run average; variance measures spread

Distribution Selection

Choose distributions based on the underlying process:

  • Bernoulli for single trials

  • Binomial for fixed trials

  • Geometric for waiting times

  • Poisson for rates

Key Principle

  • Law of Large Numbers connects theoretical expectations with observed averages

Looking Ahead

Next Lecture: Continuous Random Variables

Topics we’ll cover:

  • Probability density functions (PDFs)

  • Normal distribution

  • Exponential distribution

  • Central Limit Theorem applications

Connection: Discrete distributions often approximate continuous ones, and vice versa

Questions?

Office Hours: 11AM on Thursday (link on Canvas)

Email: nmathlouthi@ucsb.edu

Next Class: Continuous Random Variables

Resources