STAT 17: Statistical Methods for Business and Economics

Prof. Marcela Alfaro Cordoba

Statistics - UCSC

28 Apr 2026

📚 Our Journey So Far

Foundations of Statistics & Data

Statistics fundamentals: population, sample, parameters vs statistics
Sampling methods and their applications
Variable types: categorical vs numerical, discrete vs continuous
Research ethics: designing ethical experiments, identifying bias

Descriptive Statistics & Visualization

Data displays: histograms, box plots, various charts (using Google Sheets)
Central tendency: mean, median, mode + sigma notation
Measures of spread: range, variance, standard deviation
Understanding and identifying skewness in distributions

📚 Our Journey So Far

Probability Foundations

Probability terminology and rules (addition & multiplication)
Independent vs mutually exclusive events
Contingency tables: simple, marginal, and joint probability
Conditional probability P(A|B)

Meet Marcus: Campus Coffee Entrepreneur ☕

Marcus is a UCSC Economics major with entrepreneurial dreams…

His mission: Launch a student-run coffee cart at the base of campus during morning commutes

Key business questions:

How many customers will I get each hour?
What’s my expected daily revenue?
How much inventory should I stock?
What’s the probability of selling out?

The Challenge: Marcus needs to quantify uncertainty about customer counts and revenue to make smart business decisions! 🤔

Today’s goal: Use random variables and discrete distributions to model his business!

What We’ll Accomplish Today

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

✅ Define random variables and write their notation correctly (X, x, P(X=x))
✅ Understand and apply the binomial distribution to real-world counting problems
✅ Understand and apply the Poisson distribution to rate-based phenomena
✅ Calculate expected values and variances for discrete random variables
✅ Choose appropriate discrete distributions for different scenarios

What is a Random Variable? 🎲

Random Variable: A numerical value determined by the outcome of a random experiment

Key Insight: Random variables bridge the gap between random events and numerical analysis

NOT a random variable: “The next customer orders coffee” (this is an event)

IS a random variable: “The number of customers in the next hour” (assigns a number)

Marcus’s Example: Let X = number of customers during the 8-9am hour

X could be 0, 1, 2, 3, … (any non-negative integer)

Random Variable Notation 📝

Standard Notation:

X (capital letter): The random variable itself (a function)

Example: “Let X = number of customers”

x (lowercase letter): A specific value the random variable can take

Example: “x = 5 customers”

Random Variable Notation 📝

P(X = x): The probability that random variable X equals the specific value x

Example: “P(X = 5) = 0.15” means 15% chance of exactly 5 customers

Marcus’s Coffee Cart Example:

X = number of customers during 8-9am
x = 12 (a specific outcome)
P(X = 12) = probability of getting exactly 12 customers
P(X ≥ 20) = probability of getting 20 or more customers

Types of Random Variables

Discrete Random Variables

Can take on a countable number of values
Often integers
Can list all possibilities

Marcus’s Examples:

Number of customers
Number of lattes sold
Number of hours open

Continuous Random Variables

Can take on any value in an interval
Uncountable outcomes
Use intervals/ranges

Marcus’s Examples:

Time between customers
Amount of coffee in cup
Temperature of beverage

Today’s Focus: DISCRETE Random Variables (Binomial and Poisson distributions)

Probability Distribution 📊

Probability Distribution: Lists all possible values of a random variable and their associated probabilities

Marcus’s Customer Distribution (8-9am hour):

Number of Customers (x)	Probability P(X = x)
0	0.05
1	0.15
2	0.30
3	0.25
4	0.15
5	0.10
Total	1.00

Two Requirements for Valid Probability Distributions:

Each probability must be between 0 and 1: $0 \leq P(X = x) \leq 1$
All probabilities must sum to 1: $\sum P(X = x) = 1$

Expected Value (Mean) E(X) or μ 🎯

Expected Value: The long-run average value of the random variable over many repetitions

Formula:

\[E(X) = \mu = \sum [x \cdot P(X = x)]\]

“Sum of (each value × its probability)”

Calculating Marcus’s Expected Customers:

\[E(X) = 0(0.05) + 1(0.15) + 2(0.30) + 3(0.25) + 4(0.15) + 5(0.10)\]

\[E(X) = 0 + 0.15 + 0.60 + 0.75 + 0.60 + 0.50\]

\[E(X) = 2.6 \text{ customers}\]

Business Interpretation: On average, Marcus can expect 2.6 customers per hour during 8-9am. Over many days, the average will approach 2.6.

Variance and Standard Deviation σ² and σ

Variance (σ²): Measures average squared deviation from the mean

\[\sigma^2 = \sum[(x - \mu)^2 \cdot P(X = x)]\]

Standard Deviation (σ): Square root of variance (more interpretable - same units as X)

\[\sigma = \sqrt{\sigma^2}\]

Variance and Standard Deviation σ² and σ

Marcus’s Customer Variability (with μ = 2.6):

\[\sigma^2 = (0-2.6)^2(0.05) + (1-2.6)^2(0.15) + (2-2.6)^2(0.30) + ...\]

\[\sigma^2 \approx 1.64\]

\[\sigma \approx 1.28 \text{ customers}\]

Interpretation: The number of customers typically varies by about 1.28 from the average of 2.6. Most hours will see between 1-4 customers.

📊 THINK-PAIR-SHARE #1 (5 minutes)

Practice with Expected Value & Variance:

Marcus is considering a lunch hour (12-1pm) operation. Here’s his probability distribution:

Customers (x)	P(X = x)
3	0.10
4	0.20
5	0.35
6	0.25
7	0.10

Calculate:

The expected number of customers E(X)
The variance σ²
The standard deviation σ
Interpretation: If each customer spends $5 on average, what’s Marcus’s expected revenue for this hour?

Discuss with a partner, then post your calculations on Poll Everywhere!

The Binomial Distribution 🎲

When Do We Use Binomial Distribution?

The binomial distribution models situations where we have a fixed number of independent trials, each with only two possible outcomes (success/failure).

Four Conditions for Binomial Distribution (BINS):

Binary Outcomes: Each trial has exactly TWO possible outcomes
Independent Trials: One trial doesn’t affect another
Number of Trials is Fixed: We know n in advance
Same Probability: Probability of success (p) is constant

Binomial: Marcus’s Example

Scenario: Marcus offers free samples to 10 random students. Based on past data, 30% of students who try a sample make a purchase.

Let X = number of students (out of 10) who make a purchase

Check BINS conditions:

✅ B: Each student either buys or doesn’t buy
✅ I: One student’s decision doesn’t affect another’s
✅ N: Fixed at n = 10 students
✅ S: Each has p = 0.30 probability of buying

Therefore: X ~ Binomial(10, 0.30)

Binomial Distribution Notation

Standard Notation:

\[X \sim \text{Binomial}(n, p)\]

“X follows a binomial distribution with parameters n and p”

Parameters:

n: Number of independent trials (positive integer)
p: Probability of success on each trial (0 ≤ p ≤ 1)

Marcus’s Example: X ~ Binomial(10, 0.30)

n = 10 students
p = 0.30 probability of purchase
X can be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10 purchases

Binomial Probability Formula 📐

The Formula:

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

Where:

$\binom{n}{x} = \frac{n!}{x!(n-x)!}$ = “n choose x” = number of ways to arrange x successes in n trials
$p^x$ = probability of getting x successes
$(1-p)^{n-x}$ = probability of getting (n-x) failures

Example: X ~ Binomial(10, 0.30), find P(X = 3)

\[P(X = 3) = \binom{10}{3} \cdot (0.30)^3 \cdot (0.70)^7\]

\[P(X = 3) = 120 \cdot 0.027 \cdot 0.0824 \approx 0.267\]

Good news! We don’t usually calculate by hand - we use Google Sheets! 💻

Calculating Binomial Probabilities in Google Sheets 💻

BINOM.DIST Function:

=BINOM.DIST(x, n, p, cumulative)

x: number of successes you want
n: number of trials
p: probability of success on each trial
cumulative: FALSE for P(X=x), TRUE for P(X≤x)

Examples for X ~ Binomial(10, 0.30):

Probability	Formula	Result
P(X = 3)	`=BINOM.DIST(3, 10, 0.3, FALSE)`	0.2668
P(X ≤ 3)	`=BINOM.DIST(3, 10, 0.3, TRUE)`	0.6496
P(X ≥ 4)	`=1 - BINOM.DIST(3, 10, 0.3, TRUE)`	0.3504
P(2 ≤ X ≤ 5)	`=BINOM.DIST(5,10,0.3,TRUE) - BINOM.DIST(1,10,0.3,TRUE)`	0.7869

Binomial Distribution: Mean & Variance 📊

Special Formulas for Binomial!

When X ~ Binomial(n, p):

Mean (Expected Value):

\[\mu = E(X) = n \cdot p\]

Variance:

\[\sigma^2 = n \cdot p \cdot (1-p)\]

Standard Deviation:

\[\sigma = \sqrt{n \cdot p \cdot (1-p)}\]

Marcus’s Example: X ~ Binomial(10, 0.30)

$\mu = 10 \times 0.30 = 3$ purchases (expected)
$\sigma^2 = 10 \times 0.30 \times 0.70 = 2.1$
$\sigma = \sqrt{2.1} \approx 1.45$ purchases

Business Insight: Expect 3 purchases ± 1.45, so typically 2-4 purchases per 10 samples

🧘‍♀️ STRETCH BREAK

Time to recharge! (5 minutes)

Stand up and stretch 🤸‍♀️
Grab some coffee or tea ☕
Chat with neighbors 💬
Hydrate! 💧

The Poisson Distribution 🎯

When Do We Use Poisson Distribution?

The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events occur at a constant average rate and independently.

Key Characteristics:

📏 Fixed Interval: Events occur within specific time/space
⚡ Constant Rate: Events occur at consistent average rate (λ)
🎲 Independence: Events occur independently
🔢 Count Data: We count occurrences: 0, 1, 2, 3, … (no upper limit!)

Marcus’s Question: “How many customers will arrive per hour?”

From past data: average of λ = 4.5 customers per hour

Let X = number of customer arrivals in one hour → Poisson situation! ✅

Binomial vs Poisson: Key Differences 🤔

BINOMIAL

“Out of n trials, how many successes?”
Number of trials (n) is KNOWN
Parameters: n and p
Example: “Out of 10 samples, how many purchases?”
X ~ Binomial(n, p)

POISSON

“How many events in an interval?”
Time/space interval is KNOWN
Parameter: λ (average rate)
Example: “How many customers arrive per hour?”
X ~ Poisson(λ)

Key Insight:

Use Binomial when you know how many “chances” you have
Use Poisson when counting events over time/space with no fixed number of opportunities

Real-World Poisson Examples 🌍

Where Do We See Poisson Distributions?

🚗 Traffic: Number of cars passing through an intersection per minute
📞 Customer Service: Number of phone calls received at call center per hour
🏥 Healthcare: Number of patients arriving at ER per day
📧 Technology: Number of emails received per hour
🌟 Astronomy: Number of meteors visible per hour during shower
☕ Marcus’s Business: Number of customers arriving at coffee cart per hour

NOT Poisson Examples:

“Will it rain tomorrow?” (not counting events)
“How many students pass the exam?” (fixed n → use Binomial!)
“Number of heads in 20 coin flips” (fixed trials → use Binomial!)

Poisson Distribution: Notation & Formula 📐

Notation:

\[X \sim \text{Poisson}(\lambda)\]

λ (lambda) = average rate of events per interval

The Poisson Formula:

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

Where:

e ≈ 2.71828 (Euler’s number)
λ = average rate per interval
x = specific number of events (0, 1, 2, 3, …)
x! = x factorial

Example: X ~ Poisson(λ = 4.5), find P(X = 3)

\[P(X = 3) = \frac{e^{-4.5} \cdot 4.5^3}{3!} = \frac{0.0111 \times 91.125}{6} \approx 0.169\]

Calculating Poisson Probabilities in Google Sheets 💻

POISSON.DIST Function:

=POISSON.DIST(x, λ, cumulative)

x: number of events
λ: average rate (lambda)
cumulative: FALSE for P(X=x), TRUE for P(X≤x)

Examples for X ~ Poisson(4.5):

Probability	Formula	Result
P(X = 3)	`=POISSON.DIST(3, 4.5, FALSE)`	0.1687
P(X ≤ 5)	`=POISSON.DIST(5, 4.5, TRUE)`	0.7029
P(X ≥ 6)	`=1 - POISSON.DIST(5, 4.5, TRUE)`	0.2971
P(3 ≤ X ≤ 7)	`=POISSON.DIST(7,4.5,TRUE) - POISSON.DIST(2,4.5,TRUE)`	0.7116

Remember: FALSE for exact (=), TRUE for cumulative (≤)

Poisson Distribution: Mean & Variance 📊

Unique Property of Poisson!

When X ~ Poisson(λ):

Mean (Expected Value):

\[\mu = E(X) = \lambda\]

Variance:

\[\sigma^2 = \lambda\]

Standard Deviation:

\[\sigma = \sqrt{\lambda}\]

Poisson Distribution: Mean & Variance 📊

Amazing fact: For Poisson, the mean EQUALS the variance! This is a key characteristic.

Marcus’s Example: X ~ Poisson(4.5 customers per hour)

$\mu = 4.5$ customers (expected)
$\sigma^2 = 4.5$
$\sigma = \sqrt{4.5} \approx 2.12$ customers

Business Insight: Expect 4.5 customers ± 2.12, so typically 2-7 customers per hour

Changing Time Intervals with Poisson ⏰

Important Property: If events occur at rate λ per unit time, then:

For t units of time: X ~ Poisson(λt)

Marcus’s Example:

λ = 4.5 customers per hour
For 2 hours: X ~ Poisson(4.5 × 2 = 9)
For 30 minutes (0.5 hour): X ~ Poisson(4.5 × 0.5 = 2.25)
For 3 hours: X ~ Poisson(4.5 × 3 = 13.5)

Practice: What’s P(at least 10 customers in 2 hours)?

X ~ Poisson(9)

P(X ≥ 10) = 1 - P(X ≤ 9) = =1 - POISSON.DIST(9, 9, TRUE) ≈ 0.413

Choosing the Right Distribution 🎯

Decision Framework:

Use BINOMIAL when:

You have a fixed number of trials (n is known)
Each trial has two outcomes (success/failure)
Trials are independent
Same probability of success (p) each time
Question asks: “Out of n trials, how many successes?”

Use POISSON when:

Events occur over time or space (interval is known)
Events happen at a constant average rate (λ)
Events are independent
No upper limit on count
Question asks: “How many events in this interval?”

Quick Test: Can you list all possible outcomes in advance?

YES → Probably Binomial
NO (theoretically infinite) → Probably Poisson

Practice: Which Distribution? 🤔

For each scenario, identify Binomial or Poisson:

Number of customers who buy a muffin out of 20 who enter the store
- Answer: Binomial (n=20, fixed trials)

Number of students arriving at library per hour
- Answer: Poisson (counting arrivals over time)

Number of heads in 50 coin flips
- Answer: Binomial (n=50, fixed trials)

Number of typos per page in a textbook
- Answer: Poisson (counting events over space/pages)

Number of students who pass out of 100 taking an exam
- Answer: Binomial (n=100, fixed trials)

Number of emergency calls received per night
- Answer: Poisson (counting events over time)

Marcus’s Business Applications 💼

How Marcus Uses These Distributions:

Binomial Applications:

Sample promotions: “Out of n samples, how many purchases?”
Loyalty programs: “Out of n card holders, how many redeem?”
Quality control: “Out of n drinks, how many meet standards?”

Poisson Applications:

Staffing decisions: “How many customers arrive per hour?”
Inventory planning: “How many specialty orders per day?”
Equipment needs: “How many espresso orders per morning?”

Marcus’s Business Applications 💼

Combined Strategy:

Use Poisson to predict hourly customer arrivals
Use Binomial to predict purchase behavior given arrivals
Calculate expected revenue and costs
Make data-driven decisions! 📊

Calculating in Google Sheets: Summary 💻

Binomial: X ~ Binomial(n, p)

=BINOM.DIST(x, n, p, FALSE)    # P(X = x)
=BINOM.DIST(x, n, p, TRUE)     # P(X ≤ x)

Mean: =n*p
Variance: =n*p*(1-p)
Std Dev: =SQRT(n*p*(1-p))

Poisson: X ~ Poisson(λ)

=POISSON.DIST(x, lambda, FALSE)  # P(X = x)
=POISSON.DIST(x, lambda, TRUE)   # P(X ≤ x)

Mean: =lambda
Variance: =lambda
Std Dev: =SQRT(lambda)

For P(X ≥ a): Use =1 - DIST(a-1, ..., TRUE) for both!

Common Mistakes to Avoid ⚠️

Confusing n and λ:
- n is number of trials (Binomial)
- λ is average rate per interval (Poisson)
Wrong cumulative setting:
- FALSE for exact probability P(X = x)
- TRUE for cumulative P(X ≤ x)
Forgetting to adjust λ for time:
- If λ = 5 per hour, then for 3 hours use λ = 15
Using wrong distribution:
- Fixed trials → Binomial
- Counting over time/space → Poisson
Confusing “at least” and “at most”:
- P(X ≥ a) = 1 - P(X ≤ a-1)
- P(X ≤ a) = cumulative TRUE
Not checking independence assumption:
- Both distributions require independent events!

Marcus’s Success Story! 🎉

How random variables transformed his business:

✅ Inventory Management: Used Poisson to predict hourly demand → reduced waste by 40%

✅ Sampling Strategy: Used Binomial to calculate optimal sample size → 30% conversion rate

✅ Staffing Decisions: Expected value calculations → hired exactly the right number of helpers

✅ Pricing Strategy: Variance calculations → set prices to cover high-demand periods

✅ Impact: Increased profitability by 45% in first semester! Now expanding to lunch service! ☕💰

Real-World Applications Beyond Marcus 🌍

Industries Using These Distributions:

🏪 Retail: Predicting customer arrivals, purchase behavior
🏥 Healthcare: ER arrivals, patient outcomes, disease incidence
🏭 Manufacturing: Quality control, defect rates
📞 Call Centers: Call volume, service times
🚗 Transportation: Traffic flow, accident rates
📱 Technology: Server requests, bug reports, user engagement
💰 Finance: Insurance claims, loan defaults
🎓 Education: Test scores, enrollment predictions

Key Insight: Understanding random variables and probability distributions is fundamental to data-driven decision making in ANY field! 📊

Quick Knowledge Check ✅

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

Defining random variables and using correct notation ⭐⭐⭐⭐⭐
Calculating expected value and variance for discrete distributions ⭐⭐⭐⭐⭐
Identifying when to use binomial distribution (BINS conditions) ⭐⭐⭐⭐⭐
Identifying when to use Poisson distribution ⭐⭐⭐⭐⭐
Choosing the appropriate distribution for a given scenario ⭐⭐⭐⭐⭐

If you rated anything 3 or below, please visit office hours or post questions on Ed! 🤗

Practice Problems for Home 📝

Try these before next class:

Problem 1 (Binomial): A multiple-choice exam has 20 questions, each with 4 options. If a student guesses randomly on all questions:

What distribution describes the number of correct answers?
What’s the expected number of correct answers?
What’s P(passing with ≥ 12 correct)?

Problem 2 (Poisson): A website receives an average of 15 visits per hour.

What’s P(exactly 20 visits in one hour)?
What’s P(at least 10 visits in 30 minutes)?
What’s the expected number of visits in a 3-hour period?

Problem 3 (Choosing): For each scenario, identify which distribution to use and why:

Number of spam emails in your inbox per day
Number of students who complete homework out of 50 in the class
Number of cars passing a tollbooth per minute

Tips for Success 💡

Mastering discrete distributions:

Practice identifying: Read word problems and identify distribution type FIRST
Check conditions: Always verify BINS for binomial, characteristics for Poisson
Write notation: X ~ Binomial(n,p) or X ~ Poisson(λ) before calculating
Use technology: Google Sheets makes calculations fast and accurate
Interpret results: Don’t just compute - explain what it means in context!
Draw pictures: Sketch distributions to build intuition
Start early: Don’t wait until the night before the exam
Ask questions: No question is too basic - probability is tricky!

Common pitfall to avoid: Reading a problem and immediately calculating without first identifying the distribution and checking conditions!

Why This Matters 🎯

Beyond the classroom:

Understanding random variables and discrete distributions helps you:

📊 Make predictions based on uncertain data
💰 Quantify risk in business and personal decisions
🔬 Design experiments with appropriate sample sizes
📈 Interpret data from surveys and studies
🎲 Understand probability in games and gambling
🏥 Evaluate medical treatments and health risks
💼 Optimize business operations like Marcus did
🗳️ Understand polling and election forecasts

Bottom line: These tools transform vague uncertainty into quantifiable, actionable information! 🚀

Final Thoughts 💭

Remember:

Random variables turn random outcomes into numbers we can analyze
Binomial answers: “Out of n trials, how many successes?”
Poisson answers: “How many events in this time/space interval?”
Expected value and variance quantify center and spread
Google Sheets is your friend for calculations!

Marcus started with uncertainty about his coffee cart business…

Now he makes data-driven decisions using probability distributions!

You can too! 📊☕✨

See you next class for continuous distributions! 🎲

Thank you! 🎲✨

Questions? Office hours information on Canvas.

Next up: The Normal Distribution & Continuous Random Variables!

Don’t forget: Post Think-Pair-Share responses on Ed Discussion!

Appendix: Distribution Comparison Table

Feature	Binomial	Poisson
Type	Discrete	Discrete
Question	“How many out of n?”	“How many in interval?”
Parameters	n (trials), p (probability)	λ (rate)
Notation	X ~ Binomial(n, p)	X ~ Poisson(λ)
Mean	μ = np	μ = λ
Variance	σ² = np(1-p)	σ² = λ
Conditions	BINS	Fixed interval, constant rate, independence
Range	x = 0, 1, 2, …, n	x = 0, 1, 2, … (infinite)
Google Sheets	BINOM.DIST	POISSON.DIST
Example	Coin flips, surveys	Customer arrivals, defects

Appendix: Google Sheets Formulas Reference

Binomial Distribution Functions:

=BINOM.DIST(x, n, p, FALSE)           # P(X = x)
=BINOM.DIST(x, n, p, TRUE)            # P(X ≤ x)
=1 - BINOM.DIST(x-1, n, p, TRUE)      # P(X ≥ x)
=n*p                                   # Mean
=n*p*(1-p)                            # Variance
=SQRT(n*p*(1-p))                      # Standard deviation

Poisson Distribution Functions:

=POISSON.DIST(x, lambda, FALSE)       # P(X = x)
=POISSON.DIST(x, lambda, TRUE)        # P(X ≤ x)
=1 - POISSON.DIST(x-1, lambda, TRUE)  # P(X ≥ x)
=lambda                                # Mean
=lambda                                # Variance
=SQRT(lambda)                          # Standard deviation

Other Useful Functions:

=FACT(n)                              # n factorial
=COMBIN(n, x)                         # Combinations: n choose x
=EXP(x)                               # e^x

Your Random Variables Toolkit 🧰

Fundamental Concepts:

Random Variable: Numerical value from random experiment
Notation: X (variable), x (value), P(X = x) (probability)
Expected Value: E(X) = μ = long-run average
Variance: σ² = measure of spread
Standard Deviation: σ = √σ² (same units as X)

Binomial Distribution:

When: Fixed n trials, each with probability p
Notation: X ~ Binomial(n, p)
Mean: μ = np
Variance: σ² = np(1-p)
Check: BINS conditions

Poisson Distribution:

When: Count events over time/space at rate λ
Notation: X ~ Poisson(λ)
Mean: μ = λ
Variance: σ² = λ (same as mean!)
Adjust: For t intervals, use λt

Summary: Key Formulas

General Discrete Random Variable:

$E(X) = \sum [x \cdot P(X = x)]$

$\sigma^2 = \sum[(x - \mu)^2 \cdot P(X = x)]$

Binomial: X ~ Binomial(n, p):

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

$\mu = np, \quad \sigma^2 = np(1-p)$

Poisson: X ~ Poisson(λ):

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

$\mu = \lambda, \quad \sigma^2 = \lambda$

STAT 17: Statistical Methods for Business and Economics

📚 Our Journey So Far

📚 Our Journey So Far

Meet Marcus: Campus Coffee Entrepreneur ☕

What We’ll Accomplish Today

What is a Random Variable? 🎲

Random Variable Notation 📝

Random Variable Notation 📝

Types of Random Variables

Probability Distribution 📊

Expected Value (Mean) E(X) or μ 🎯

Variance and Standard Deviation σ² and σ

Variance and Standard Deviation σ² and σ

📊 THINK-PAIR-SHARE #1 (5 minutes)

Share your answers in Poll Everywhere!

The Binomial Distribution 🎲

Binomial: Marcus’s Example

Binomial Distribution Notation

Binomial Probability Formula 📐

Calculating Binomial Probabilities in Google Sheets 💻

Binomial Distribution: Mean & Variance 📊

📊 THINK-PAIR-SHARE #2 (7 minutes)

🧘‍♀️ STRETCH BREAK

Time to recharge! (5 minutes)

The Poisson Distribution 🎯

Binomial vs Poisson: Key Differences 🤔

Real-World Poisson Examples 🌍

Poisson Distribution: Notation & Formula 📐

Calculating Poisson Probabilities in Google Sheets 💻

Poisson Distribution: Mean & Variance 📊

Poisson Distribution: Mean & Variance 📊

Changing Time Intervals with Poisson ⏰

📊 THINK-PAIR-SHARE #3 (7 minutes)

Choosing the Right Distribution 🎯

Practice: Which Distribution? 🤔

Marcus’s Business Applications 💼

Marcus’s Business Applications 💼

Calculating in Google Sheets: Summary 💻

Common Mistakes to Avoid ⚠️

Marcus’s Success Story! 🎉

Real-World Applications Beyond Marcus 🌍

Quick Knowledge Check ✅

Practice Problems for Home 📝

Tips for Success 💡

Why This Matters 🎯

Final Thoughts 💭

Thank you! 🎲✨

Appendix: Distribution Comparison Table

Appendix: Google Sheets Formulas Reference

Your Random Variables Toolkit 🧰

Fundamental Concepts:

Binomial Distribution:

Poisson Distribution:

Summary: Key Formulas