STAT 17: Statistical Methods for Business and Economics

Prof. Marcela Alfaro Cordoba

Statistics - UCSC

28 Apr 2026

Quick Review: Where We Left Off

Binomial Distribution: X ~ Binomial(n, p)

  • When to use: Fixed number of independent trials (n), each with probability p
  • BINS conditions: Binary, Independent, Number fixed, Same probability
  • Mean: μ = np
  • Variance: σ² = np(1-p)
  • Google Sheets: =BINOM.DIST(x, n, p, FALSE/TRUE)

Quick Review: Where We Left Off

Binomial Distribution

For X ~ Binomial(n, p):

\[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, 2, \ldots, n\]

Where:

  • \(\binom{n}{x} = \frac{n!}{x!(n-x)!}\) = binomial coefficient
  • n = number of independent trials
  • p = probability of success on each trial
  • x = number of successes
  • (1-p) = probability of failure

Quick Review: Where We Left Off

Poisson Distribution: X ~ Poisson(λ)

  • When to use: Counting events over time/space at constant rate λ
  • Mean = Variance: μ = σ² = λ
  • Time adjustment: For t intervals, use λt
  • Google Sheets: =POISSON.DIST(x, lambda, FALSE/TRUE)

Quick Review: Where We Left Off

Poisson Distribution

For X ~ Poisson(λ):

\[P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = 0, 1, 2, \ldots\]

Where:

  • e ≈ 2.71828 (Euler’s number)
  • λ = average rate of events per interval
  • x = number of events
  • x! = x factorial

What We’ll Accomplish Today

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

  • Calculate ALL types of probabilities with discrete distributions
  • Choose appropriate distributions for given real-world scenarios
  • Calculate expected values and variances for functions of random variables
  • Understand continuous random variables and their key differences
  • Work with the uniform distribution (if there’s time)

Mastering Probability Calculations

For ANY discrete random variable, we need to calculate:

  • P(X = x) - exactly x
  • P(X ≤ x) - x or fewer
  • P(X < x) - fewer than x
  • P(X ≥ x) - x or more
  • P(X > x) - more than x
  • P(a ≤ X ≤ b) - between a and b

Key Insight: Master the relationships between these to solve ANY probability problem!

Exact Probability: P(X = x)

Meaning: The probability that X takes on exactly the value x

Google Sheets Formula:

=BINOM.DIST(x, n, p, FALSE)
=POISSON.DIST(x, lambda, FALSE)

Note: Use FALSE for exact probability!

Example: X = number of orders in a coffee shop from 4pm - 5pm, X ~ Poisson(5), find P(X = 7)

\[P(X = 7) = \frac{e^{-\lambda} \lambda^x}{x!} = \frac{e^{-5} 5^7}{7!} = 0.1044\]

=POISSON.DIST(7, 5, FALSE) ≈ 0.1044

Interpretation: About 10.44% chance of exactly 7 orders

Exact Probability: P(X = x)

Meaning: The probability that X takes on exactly the value x

Google Sheets Formula:

=BINOM.DIST(x, n, p, FALSE)
=POISSON.DIST(x, lambda, FALSE)

Note: Use FALSE for exact probability!

Example: X = number of people who order a latte out of 20 clients. X ~ Binomial(n=20, p = 0.4), find P(X = 7)

\[P(X = 7) = \binom{n}{x} p^x (1-p)^{n-x} = \binom{20}{7} 0.4^7 (1-0.4)^{20-7} = 0.1659\]

BINOM.DIST(7, 20, 0.4, FALSE) ≈ 0.1659

Interpretation: About 16.59% probability of getting exactly 7 orders of latte among 20 clients.

Cumulative: P(X ≤ x) At Most

Meaning: Probability that X is less than or equal to x

Google Sheets Formula:

=BINOM.DIST(x, n, p, TRUE)
=POISSON.DIST(x, lambda, TRUE)

Note: Use TRUE for cumulative probability!

Example: X ~ Binomial(20, 0.05), find P(X ≤ 2)

\[P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)\] \[ = \binom{20}{0} 0.05^0 (1-0.05)^{20-0} + \binom{20}{1} 0.05^1 (1-0.05)^{20-1} + \] \[\binom{20}{2} 0.05^2 (1-0.05)^{20-2}= 0.9245\]

=BINOM.DIST(2, 20, 0.05, TRUE) ≈ 0.9245

P(X < x) Less Than

Critical Distinction: Less than x does NOT include x itself!

The Formula: P(X < x) = P(X ≤ x-1)

Google Sheets:

=BINOM.DIST(x-1, n, p, TRUE)
=POISSON.DIST(x-1, lambda, TRUE)

Example: X ~ Poisson(8), find P(X < 5)

\[P(X < 5) = \sum_{i = 0, n =5} P(X = i)\] \[P(X = 0) = \frac{e^{-8} \cdot 8^0}{0!} = \frac{0.00033546 \cdot 1}{1} = 0.00033546\]

\[P(X = 1) = \frac{e^{-8} \cdot 8^1}{1!} = \frac{0.00033546 \cdot 8}{1} = 0.00268368\]

\[P(X = 2) = \frac{e^{-8} \cdot 8^2}{2!} = \frac{0.00033546 \cdot 64}{2} = 0.01073472\]

\[P(X = 3) = \frac{e^{-8} \cdot 8^3}{3!} = \frac{0.00033546 \cdot 512}{6} = 0.02862592\]

\[P(X = 4) = \frac{e^{-8} \cdot 8^4}{4!} = \frac{0.00033546 \cdot 4096}{24} = 0.05725184\]

\[P(X < 5) = 0.00033546 + 0.00268368 + 0.01073472 + 0.02862592 + 0.05725184\]

\[P(X < 5) = 0.09963162\]

Answer: P(X < 5) ≈ 0.0996 or 9.96%

=POISSON.DIST(4, 8, TRUE) ≈ 0.0996

P(X ≥ x) At Least - Complement Rule

The Formula: P(X ≥ x) = 1 - P(X ≤ x-1)

Google Sheets:

=1-BINOM.DIST(x-1, n, p, TRUE)
=1-POISSON.DIST(x-1, lambda, TRUE)

Example: X ~ Binomial(50, 0.30), find P(X ≥ 20)

P(X ≥ 20) = 1 - P(X < 19)

=1-BINOM.DIST(19, 50, 0.30, TRUE) ≈ 0.0848

P(X > x) More Than

The Formula: P(X > x) = 1 - P(X ≤ x)

Compare: - P(X ≥ x) = 1 - P(X ≤ x-1) includes x - P(X > x) = 1 - P(X ≤ x) excludes x

Example: X ~ Poisson(12), find P(X > 15)

=1-POISSON.DIST(15, 12, TRUE) ≈ 0.1556

Range Probabilities

The Master Formula: P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1)

All Four Range Types:

  • P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a-1)
  • P(a < X ≤ b) = P(X ≤ b) - P(X ≤ a)
  • P(a ≤ X < b) = P(X ≤ b-1) - P(X ≤ a-1)
  • P(a < X < b) = P(X ≤ b-1) - P(X ≤ a)

Example: X ~ Poisson(6), find P(4 ≤ X ≤ 8)

=POISSON.DIST(8,6,TRUE) - POISSON.DIST(3,6,TRUE) ≈ 0.6960

THINK-PAIR-SHARE 1 (7 minutes)

Probability Practice Problems

A call center receives an average of 10 calls per hour. Let X ~ Poisson(10)

Calculate:

  1. P(X = 12) - exactly 12 calls

  2. P(X ≤ 8) - at most 8 calls

  3. P(X < 7) - fewer than 7 calls

  4. P(X ≥ 15) - at least 15 calls

  5. P(X > 13) - more than 13 calls

  6. P(8 ≤ X ≤ 12) - between 8 and 12 calls

Write Google Sheets formulas AND calculate answers! Post on Ed Discussion with your partner’s name!

Choosing the Right Distribution

Use BINOMIAL when:

  • Fixed number of trials (n is known)

  • Each trial: success or failure

  • Trials are independent

  • Same probability p each time

Use POISSON when:

  • Counting events over time/space

  • Events happen at constant rate

  • Events are independent

  • No upper limit on count

Quick Test: Can you list all outcomes? YES = Binomial, NO = Poisson

Practice: Which Distribution?

  1. Website gets 200 visitors, 5% buy - how many purchases?

  2. How many typos in 50-page document?

  3. Out of 100 coin flips, how many heads?

  4. How many emails arrive per day?

  5. Inspect 50 products, how many defective?

Share your answers in Poll Everywhere!

Choose the right answer according to the question number.

STRETCH BREAK (5 minutes)

Time to recharge!

  • Stand up and stretch
  • Grab water or coffee
  • Chat with neighbors
  • Rest your eyes

Functions of Random Variables

New Concept: What if we transform a random variable?

Linear Transformation: Y = aX + b

Example: Coffee shop with X ~ Poisson(8) customers per hour Each customer spends 5 dollars Revenue: Y = 5X

How do we find E(Y) and Var(Y)?

Expected Value of Linear Functions

The Rules:

E(aX + b) = aE(X) + b

Special cases:

  • E(aX) = aE(X)

  • E(X + b) = E(X) + b

  • E(c) = c

Example: X ~ Poisson(8), Revenue Y = 5X

E(Y) = 5 × E(X) = 5 × 8 = 40 dollars per hour

With Fixed Cost: Z = 3X + 50, E(X) = 20

E(Z) = 3 × 20 + 50 = 110 dollars

Variance of Linear Functions

The Rules:

Var(aX + b) = a² Var(X)

IMPORTANT: Adding a constant does not change variance!

Special cases:

  • Var(aX) = a² Var(X)

  • Var(X + b) = Var(X)

  • Var(c) = 0

Example: X ~ Poisson(8), Revenue Y = 5X

Var(Y) = 25 × 8 = 200

SD(Y) = sqrt(200) ≈ 14.14 dollars

Complete Example: Delivery Service

X ~ Poisson(12) deliveries per day E(X) = 12, Var(X) = 12

  • Daily revenue: R = 8X + 25

Calculate E(R):

E(R) = 8 × 12 + 25 = 121 dollars per day

Calculate Var(R):

Var(R) = 64 × 12 = 768

SD(R) = sqrt(768) ≈ 27.71 dollars

THINK-PAIR-SHARE 2 (7 minutes)

Functions of Random Variables Practice

Ride-sharing driver: X ~ Poisson(15) rides per day

Earnings model: E = 12X + 40

Calculate:

  1. E(X) and Var(X)

  2. E(E) - expected daily earnings

  3. Var(E) - variance of earnings

  4. SD(E) - standard deviation

  5. Interpretation: What do these tell the driver?

Show all work! Post on Ed Discussion with partner’s name!

Transitioning to Continuous Random Variables

Discrete:

  • Countable values (0, 1, 2, …)
  • Can list all outcomes
  • P(X = x) makes sense
  • Examples: customers, coin flips

But what about:

  • Time until next customer
  • Height of student
  • Temperature tomorrow
  • Exact amount spent
  • Weight of product

These take ANY value in an interval - CONTINUOUS!

Continuous Random Variables

Definition: Can take any value within an interval

DISCRETE vs CONTINUOUS:

  • Discrete: Countable, gaps, P(X=x) works, sum probabilities
  • Continuous: Uncountable, no gaps, P(X=x)=0, integrate areas

Critical Difference: For continuous, P(X = any specific value) = 0

We can only calculate P(a ≤ X ≤ b) over intervals!

Why P(X = x) = 0 for Continuous?

Intuition: Infinite precision means infinitely many values

Example: Height exactly 170 cm?

Not 169.9 or 170.1, but exactly 170.000000… cm?

With infinite precision, probability is 0!

But we CAN ask:

  • P(169 ≤ Height ≤ 171) = positive

  • P(Height ≤ 170) = positive

  • P(Height > 165) = positive

Key Consequence: For continuous, P(X ≤ a) = P(X < a)

Probability Density Function (PDF)

For discrete: Probability mass function (PMF)

For continuous: Probability density function (PDF), f(x)

Key Properties:

  1. f(x) ≥ 0 for all x

  2. Total area under f(x) = 1

  3. P(a ≤ X ≤ b) = area under f(x) from a to b

  4. f(x) itself is NOT a probability

For continuous: P(a ≤ X ≤ b) = area under PDF

The Uniform Distribution

Our First Continuous Distribution!

Uniform: All values in interval equally likely

Notation: X ~ Uniform(a, b)

PDF: f(x) = 1/(b-a) for a ≤ x ≤ b, otherwise 0

The PDF is constant (flat) over the interval!

Example: Bus arrives between 8:00 and 8:20

X ~ Uniform(0, 20) minutes after 8:00

f(x) = 1/20 for 0 ≤ x ≤ 20

Uniform: Mean and Variance

For X ~ Uniform(a, b):

Mean: E(X) = (a + b)/2

Variance: Var(X) = (b-a)²/12

Standard Deviation: SD(X) = (b-a)/sqrt(12)

Example: X ~ Uniform(0, 20)

  • E(X) = 10 minutes

  • Var(X) = 400/12 ≈ 33.33

  • SD(X) ≈ 5.77 minutes

Calculating Probabilities: Uniform

For X ~ Uniform(a, b):

P(c ≤ X ≤ d) = (d - c)/(b - a)

This is area of rectangle!

Height = 1/(b-a), Width = (d-c), Area = (d-c)/(b-a)

Example: X ~ Uniform(0, 20)

P(5 ≤ X ≤ 15) = (15-5)/(20-0) = 10/20 = 0.50

50% chance bus arrives in that window

More Uniform Examples

X ~ Uniform(0, 20)

  1. P(X ≤ 10) = 10/20 = 0.50

  2. P(X > 15) = 5/20 = 0.25

  3. P(X < 5 or X > 18) = 5/20 + 2/20 = 0.35

  4. P(8 ≤ X ≤ 12) = 4/20 = 0.20

Google Sheets: Uniform Distribution

No built-in function, but easy formulas!

For X ~ Uniform(a, b):

Mean: =(a+b)/2
Variance: =(b-a)^2/12
Std Dev: =(b-a)/SQRT(12)
P(c ≤ X ≤ d): =(d-c)/(b-a)
P(X ≤ c): =(c-a)/(b-a)
P(X ≥ d): =(b-d)/(b-a)

Example: X ~ Uniform(5, 15)

Mean: =(5+15)/2 = 10

P(7 ≤ X ≤ 12): =(12-7)/(15-5) = 0.5

Real-World Uniform Applications

When do we see uniform distributions?

  1. Random number generators
  2. Round-off errors in measurements
  3. Wait times (bus every 30 minutes)
  4. Angle measurements
  5. Starting assumptions with no information

THINK-PAIR-SHARE 3 (7 minutes)

Uniform Distribution Practice

Customer arrives uniformly between 7:15 and 7:45

X ~ Uniform(15, 45) minutes after 7:00

Calculate:

  1. E(X) - expected arrival time

  2. Var(X) and SD(X)

  3. P(X ≤ 30) - arrive by 7:30

  4. P(X > 40) - arrive after 7:40

  5. P(20 ≤ X ≤ 35) - between 7:20 and 7:35

  6. P(within 5 minutes of 7:30)

Share your answers in Poll Everywhere!

  1. P(within 5 minutes of 7:30)

Comparing Discrete vs Continuous

Feature Discrete Continuous
Values Countable Uncountable
Function PMF PDF
P(X=x) Can be positive Always 0
Probabilities Sum Integrate
Total Prob Sum = 1 Integral = 1
Examples Binomial, Poisson Uniform, Normal
Less than vs Less than or equal Different Same

Key: Continuous needs intervals, not exact values!

Summary: What We Covered Today

Part 1: Mastering Discrete Probabilities

  • All probability types

  • Choosing distributions

  • Complement rule

Part 2: Functions of Random Variables

  • E(aX + b) = aE(X) + b

  • Var(aX + b) = a² Var(X)

  • Real-world applications

Part 3: Continuous Distributions

  • P(X = x) = 0

  • PDF and area under curve

  • Uniform distribution

Quick Knowledge Check ✅

Rate your confidence (1-5) on Ed Discussion:

  1. Calculating all probability types by hand ⭐⭐⭐⭐⭐
  2. Calculating all probability types using Google Sheets ⭐⭐⭐⭐⭐
  3. Choosing appropriate distribution for scenarios ⭐⭐⭐⭐⭐
  4. Calculating E(aX + b) and Var(aX + b) ⭐⭐⭐⭐⭐
  5. Understanding discrete vs continuous ⭐⭐⭐⭐⭐
  6. Working with uniform distribution ⭐⭐⭐⭐⭐

If you rated anything 3 or below, visit office hours!

Thank you! 📊✨

Questions? I have office hours right after class today!

Next up: The Normal Distribution

Remember:

  • Post Think-Pair-Share on Ed Discussion and Poll Everywhere
  • Rate your confidence
  • Start midterm practice problems early (HW4 will be posted today)