Lecture 7: Random Variables and Named Distributions

STAT 7 - Statistical Methods for the Biological, Environmental & Health Sciences

Prof. Marcela Alfaro Córdoba

Statistics - UCSC

10 Mar 2026

Welcome! Week 4 Overview

Week Topics Readings Deliverables
Four Random Variables & Distributions Sections 4.1, 4.2, 4.3 Complete HW3 and Practice Exam (HW4)
Attend 2 lectures, 1 DSA

Recap: What We’ve Covered

  • Statistics fundamentals & data types
  • Descriptive statistics & data visualization
  • Exploratory data analysis
  • Probability basics
  • Conditional probability
  • Bayes’ Theorem

Today: We connect probability to real-world phenomena through random variables!

Why Do We Need Probability?

Remember Week 1:

  • We want to make inferences about populations
  • We can only observe samples
  • Need to quantify uncertainty

This Week:

  • How do variables behave randomly?
  • What patterns do they follow?
  • How can we model uncertainty mathematically?

Quick Review: Bayes’ Theorem

Remember the general form:

\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]

Example: Medical Testing

  • Disease prevalence: 1% of population
  • Test sensitivity: 95% (true positive rate)
  • Test specificity: 90% (true negative rate)

If you test positive, what’s the probability you have the disease?

Bayes’ Theorem Example Solution

Let D = has disease, T+ = tests positive

\[P(D|T+) = \frac{P(T+|D) \cdot P(D)}{P(T+)}\]

  • \(P(D) = 0.01\)
  • \(P(T+|D) = 0.95\)
  • \(P(T+) = P(T+|D) \cdot P(D) + P(T+|D^c) \cdot P(D^c)\)
  • \(P(T+) = 0.95(0.01) + 0.10(0.99) = 0.1085\)

\[P(D|T+) = \frac{0.95 \times 0.01}{0.1085} \approx 0.088\]

Only 8.8% chance! Base rates matter enormously.

Today’s Topics

  1. Random Variables
  2. Expected Value & Variance
  3. Named Probability Distributions
  4. Binomial Distribution

Random Variables

What is a Random Variable?

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

Example:

Flip a coin 3 times

Let X = number of heads

X maps outcomes to numbers

Random Variables: Examples

Which of these are random variables?

  1. The number of students absent tomorrow
  2. Tomorrow’s high temperature in °C
  3. The color of the next car you see
  4. Time until your next email arrives
  5. Whether it rains tomorrow (Yes/No)

Types of Random Variables

Discrete

  • Countable values
  • Often whole numbers
  • Gaps between values

Examples:

  • Number of heads in 10 flips
  • Number of eggs in a nest
  • Number of students in class

Continuous

  • Uncountable values
  • Any value in an interval
  • No gaps

Examples:

  • Height of a person
  • Time to complete a task
  • Temperature
  • Rainfall amount

Classification Exercise

Classify each as discrete or continuous:

  1. Number of heads in three tosses of a coin → Discrete
  2. Time to finish a marathon (in seconds) → Continuous
  3. Number of eggs in an eagle’s nest → Discrete
  4. Body mass of whales (in kg) → Continuous
  5. Number of accidents at an intersection in a year
  6. Daily rainfall amount (in inches)
  7. Number of people with type O blood in a sample
  8. Plant height (in meters)

Probability Distributions

For a discrete random variable X:

Probability Distribution = List of all possible values and their probabilities

Requirements:

  1. \(0 \leq P(X = x) \leq 1\) for all x
  2. \(\sum_{all\ x} P(X = x) = 1\)

Example: Coin Flips

Flip a fair coin 3 times. Let X = number of heads.

Sample Space:

TTT, TTH, THT, HTT, THH, HTH, HHT, HHH

Probability Distribution:

X 0 1 2 3
P(X) 1/8 3/8 3/8 1/8

Visualization:

Expected Value (Mean)

Expected Value = long-run average value of the random variable

For discrete X:

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

Example (coin flips):

\[E(X) = 0 \cdot \frac{1}{8} + 1 \cdot \frac{3}{8} + 2 \cdot \frac{3}{8} + 3 \cdot \frac{1}{8} = 1.5\]

Variance and Standard Deviation

Variance = average squared deviation from the mean

\[Var(X) = \sigma^2 = \sum_{all\ x} (x - \mu)^2 \cdot P(X = x)\]

Standard Deviation:

\[SD(X) = \sigma = \sqrt{Var(X)}\]

Measures the spread or variability of the distribution

Calculate Together

For our coin flip example (X = number of heads in 3 flips):

We found: \(E(X) = 1.5\)

Calculate Variance:

\[\begin{align} Var(X) &= (0-1.5)^2 \cdot \frac{1}{8} + (1-1.5)^2 \cdot \frac{3}{8} \\ &\quad + (2-1.5)^2 \cdot \frac{3}{8} + (3-1.5)^2 \cdot \frac{1}{8} \\ &= 2.25 \cdot \frac{1}{8} + 0.25 \cdot \frac{3}{8} + 0.25 \cdot \frac{3}{8} + 2.25 \cdot \frac{1}{8} \\ &= 0.75 \end{align}\]

Standard Deviation: \(SD(X) = \sqrt{0.75} \approx 0.87\)

Named Probability Distributions

Why Named Distributions?

Rather than deriving probability distributions from scratch every time, statisticians have identified common patterns:

  • Binomial: Success/failure trials
  • Normal: Bell-curved continuous data
  • Poisson: Rare events over time/space
  • Exponential: Time until an event
  • And many more!

Each has a specific formula and properties.

Family of Distributions

Today we focus on Binomial and Normal

Binomial Distribution

The Binomial Setting

Four conditions must be met:

  1. Fixed number of trials (n)
  2. Each trial has two outcomes (success/failure)
  3. Trials are independent
  4. Probability of success (p) is constant for each trial

Notation: \(X \sim Binomial(n, p)\) or \(X \sim Bin(n, p)\)

X = number of successes in n trials

Binomial Example: Plant Germination

A botanist plants 50 seeds, each with 80% germination probability.

Check the conditions:

  1. Fixed trials? Yes, n = 50 seeds
  2. Two outcomes? Yes, germinates or doesn’t
  3. Independent? Yes, separate containers
  4. Constant probability? Yes, p = 0.80 for each

✓ This is a Binomial setting!

Binomial Formula

Probability of exactly k successes:

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

Where:

  • \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is the binomial coefficient
  • \(n\) = number of trials
  • \(k\) = number of successes
  • \(p\) = probability of success

Mean and Variance:

\[E(X) = np \qquad Var(X) = np(1-p)\]

Binomial Calculation Example

Setup: 10 seeds, p = 0.8

What’s the probability exactly 10 seeds germinate?

\[P(X = 10) = \binom{10}{10} (0.8)^{10}(0.2)^{0}\]

\[= 1 \times 0.107 \times 1 = 0.107\]

About 10.7% chance all 10 germinate

Using Google Sheets for Binomial

We don’t calculate by hand! Use BINOM.DIST()

Syntax:

=BINOM.DIST(k, n, p, cumulative)
  • k = number of successes
  • n = number of trials
  • p = probability of success
  • cumulative = FALSE for P(X = k), TRUE for P(X ≤ k)

Example: =BINOM.DIST(10, 10, 0.8, FALSE) gives 0.107

Other option: Stat Crunch

Practice Problem: Plant Seeds

A botanist plants 10 seeds, each with 80% germination probability.

Calculate:

  1. P(exactly 10 germinate)
    • =BINOM.DIST(10, 10, 0.8, FALSE) = 0.107
  2. P(fewer than 5 germinate)
    • =BINOM.DIST(4, 10, 0.8, TRUE) = 0.0064
  3. P(more than 6 germinate)
    • =1 - BINOM.DIST(6, 10, 0.8, TRUE) = 0.879
  4. Expected number that germinate
    • \(E(X) = np = 10 \times 0.8 = 8\)

Binomial Distribution Shape

Shape depends on p: symmetric when p=0.5, skewed otherwise

Stretch Break!

Let’s take 2 minutes to stretch!

Looking Ahead: Thursday

We’ll cover:

  • Everything about the Normal distribution
  • Normal approximation to binomial (when is it appropriate?)
  • Sampling distributions
  • Central Limit Theorem
  • Parameters vs. Statistics

Be ready to connect probability to inference!

Exit Ticket

Before you leave, please complete:

  1. One question you still have about random variables or distributions

  2. Complete by Friday:

    • HW 3 on Canvas

  3. Complete during Discussion Section:

    • DSA 4

  4. Study for Midterm (next week!)

Office hours available - check Ed Discussion for times

Summary

  • Random variables assign numbers to outcomes
  • Discrete vs continuous
  • Expected value, variance, standard deviation
  • Binomial: Fixed trials, constant probability
  • Use Google Sheets to calculate probabilities!

Questions?