Probability Distributions

🎲 The Magic of Probability Distributions in R

The Story of Probability: Your Crystal Ball for Data

Imagine you have a magic bag of marbles. Some are red, some blue, some green. If you close your eyes and pick one, can you guess which color you’ll get? That’s what probability helps you figure out!

Probability distributions are like recipe cards that tell you how likely different outcomes are. Just like a recipe tells you how many eggs and how much flour you need, a probability distribution tells you how often different things happen.

In R, we have special tools to work with these “recipes.” Let’s explore 8 amazing distributions!

🔔 Normal Distribution: The Bell Curve

What Is It?

Think of measuring the heights of everyone in your school. Most kids are around the middle height. Very few are super tall or super short. If you draw this on paper, it looks like a bell!

graph TD
    A["Few very short"] --> B["Some short"]
    B --> C["MOST ARE AVERAGE"]
    C --> D["Some tall"]
    D --> E["Few very tall"]

Real Life Examples

• Heights of people
• Test scores in a big class
• How long it takes to run to the bus stop

R Commands

# Create a bell curve
dnorm(x, mean, sd)  # Height of curve
pnorm(x, mean, sd)  # Area under curve
qnorm(p, mean, sd)  # Find the x value
rnorm(n, mean, sd)  # Random numbers

Simple Example

# 100 test scores
# Average is 75, spread is 10
scores <- rnorm(100, mean=75, sd=10)
hist(scores)

What the numbers mean:

• mean = 75 → The middle point
• sd = 10 → How spread out the scores are

🎯 Binomial Distribution: Yes or No Game

What Is It?

Flip a coin 10 times. How many heads will you get? That’s binomial! It counts successes when you try something multiple times.

Think of it like this: You shoot 5 basketball free throws. Each shot is either IN or MISS. The binomial tells you the chance of getting exactly 3 baskets!

The Magic Formula in R

# How likely is exactly k successes?
dbinom(k, size, prob)

# Probability of k or fewer
pbinom(k, size, prob)

# Generate random results
rbinom(n, size, prob)

Real Example

# Flip a fair coin 10 times
# What's the chance of exactly 6 heads?
dbinom(6, size=10, prob=0.5)
# Answer: about 20.5%

# Simulate 100 experiments
flips <- rbinom(100, size=10, prob=0.5)

Key parts:

• size = How many tries
• prob = Chance of success each time

🌟 Poisson Distribution: Counting Rare Events

What Is It?

How many shooting stars will you see tonight? How many customers enter a shop per hour? How many emails land in your inbox?

Poisson counts rare things that happen over time or space.

graph TD
    A["Count events"] --> B["In a fixed time"]
    A --> C["Events are random"]
    A --> D["Events are independent"]

Why It’s Cool

If you know that, on average, 3 buses arrive per hour, Poisson tells you the chance of seeing exactly 5 buses!

R Commands

# Chance of exactly k events
dpois(k, lambda)

# Chance of k or fewer events
ppois(k, lambda)

# Random counts
rpois(n, lambda)

Example

# Average 4 phone calls per hour
# Chance of exactly 6 calls?
dpois(6, lambda=4)
# Answer: about 10.4%

# Simulate a week of hourly calls
calls <- rpois(168, lambda=4)

Remember: lambda = average number of events

📏 Uniform Distribution: Equal Chances

What Is It?

Roll a fair die. Every number (1, 2, 3, 4, 5, 6) has the same chance! That’s uniform distribution - everyone gets an equal slice of the pie!

Two Types

R Commands

# Continuous uniform
dunif(x, min, max)  # Height
punif(x, min, max)  # Area
runif(n, min, max)  # Random numbers

# Example: Random number 0-100
runif(1, min=0, max=100)

Simple Example

# Spinner that lands anywhere from 0 to 360
angle <- runif(1, min=0, max=360)

# 50 random ages between 18 and 65
ages <- runif(50, min=18, max=65)

⏱️ Exponential Distribution: Waiting Time

What Is It?

How long until the next bus arrives? How long does a light bulb last? Exponential distribution tells you about waiting times between events!

The Story

Imagine waiting at a bus stop. Some waits are short (lucky!), some are long (ugh!). Most waits are short, but occasionally you wait forever. That pattern is exponential!

R Commands

# The shape of the curve
dexp(x, rate)

# Chance of waiting less than x
pexp(x, rate)

# Random wait times
rexp(n, rate)

Example

# Average 2 customers per minute
# = wait time rate of 2

# Chance of waiting less than 30 sec?
pexp(0.5, rate=2)
# Answer: about 63%

# Simulate 100 wait times
waits <- rexp(100, rate=2)

Key: rate = how often events happen (per unit time)

📊 t-Distribution: The Small Sample Hero

What Is It?

What if you only have a few data points? The normal distribution doesn’t work well. Enter the t-distribution - a superhero for small samples!

graph TD
    A["Small sample?"] --> B["Use t-distribution!"]
    B --> C["Wider tails"]
    C --> D["More uncertainty"]
    D --> E["Safer conclusions"]

How It’s Different From Normal

• Fatter tails = More extreme values possible
• With more data, it becomes normal
• Uses degrees of freedom (df)

R Commands

# Shape of the curve
dt(x, df)

# Area under curve
pt(x, df)

# Random t-values
rt(n, df)

Example

# Only 10 students in sample
# degrees of freedom = 10 - 1 = 9

# Generate t-statistics
t_values <- rt(1000, df=9)

# Compare to normal
normal_values <- rnorm(1000)

Remember: df = sample size minus 1

📈 Chi-Square Distribution: Testing Categories

What Is It?

Do boys and girls like the same ice cream flavors equally? Is a die fair? Chi-square helps you test if things match what you expected!

The Magic

It measures: How far is reality from expectation?

• If chi-square is small → Reality matches expectation
• If chi-square is big → Something unexpected is happening!

R Commands

# Curve shape
dchisq(x, df)

# Probability
pchisq(x, df)

# Random values
rchisq(n, df)

Example

# Testing if a die is fair
# 6 categories, so df = 6 - 1 = 5

# Get critical value (95% confidence)
qchisq(0.95, df=5)
# Answer: 11.07

# Simulate chi-square values
chi_vals <- rchisq(1000, df=5)

When to use: Comparing observed counts vs expected counts

⚖️ F-Distribution: Comparing Variances

What Is It?

Are two groups equally spread out? The F-distribution compares the variability of two groups!

Real Life Example

Imagine two teachers give the same test:

• Class A: Scores are all close together
• Class B: Scores are spread all over

F-distribution tells you if this difference is real or just random luck!

graph TD
    A["Group 1 variance"] --> C["F = ratio"]
    B["Group 2 variance"] --> C
    C --> D["Compare to F-distribution"]
    D --> E{Significant?}

R Commands

# Curve shape
df(x, df1, df2)

# Probability
pf(x, df1, df2)

# Random F-values
rf(n, df1, df2)

Example

# Comparing two groups
# Group 1: 10 items (df1 = 9)
# Group 2: 15 items (df2 = 14)

# Critical F-value (95%)
qf(0.95, df1=9, df2=14)
# Answer: 2.65

# Simulate F-ratios
f_vals <- rf(1000, df1=9, df2=14)

Two df’s: One for each group (sample size minus 1)

🗺️ Quick Reference Map

🎮 The R Function Pattern

All distributions in R follow the same pattern:

This pattern works for every distribution! Once you learn it, you’ve learned them all!

🌈 The Big Picture

Think of distributions as different shapes of uncertainty:

1. Normal → The friendly bell (most common)
2. Binomial → The yes/no counter
3. Poisson → The rare event tracker
4. Uniform → The fair dealer
5. Exponential → The wait timer
6. t → The small sample protector
7. Chi-Square → The category checker
8. F → The variance comparer

Each one is a tool in your data scientist toolbox. Pick the right tool for the job, and R makes it easy!

🚀 You’ve Got This!

You now understand 8 powerful probability distributions! These aren’t just math concepts - they’re the building blocks of data science, machine learning, and understanding the world around you.

Next time you see a bell curve, count successes, or wait for something to happen, you’ll know exactly which distribution is at work!

Remember: Every d, p, q, and r function in R follows the same pattern. Master one, master them all!