Normal Distribution (Gaussian)

• Origins: Developed by Carl Friedrich Gauss
• Also known as the “bell curve”
• Arose from the study of measurement errors in astronomy

Common Occurrences Normal Distribution in Nature

• Human height and weight
• IQ scores
• Measurement errors
• Blood pressure readings

Mathematical Formula

\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Why It Occurs

• Central Limit Theorem
• Result of many small, independent effects
• Most biological variations

Poisson Distribution

• Origins: Developed by Siméon Denis Poisson
• Originally used to model rare events
• Named after studying “small numbers”

Mathematical Formula

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

Common Occurrences Poisson Distribution in Nature

• Number of:
  • Mutations in DNA sequence
  • Customer arrivals
  • Defects in manufacturing
  • Rare disease cases
• Traffic flow
• Radioactive decay events

Gamma Distribution

• Extension of factorial to real numbers
• Fundamental in statistical theory
• Related to waiting times

Common Occurrences of Gamma Distribution in Nature

• Waiting times
• Rainfall amounts
• Insurance claims
• Service times
• Life spans of devices

Mathematical Formula

\[ f(x) = \frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)} \]

Lognormal Distribution

• Transformation of normal distribution
• Product of many small independent factors
• Discovered through study of growth rates

Mathematical Formula

\[ f(x) = \frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac{(\ln x-\mu)^2}{2\sigma^2}} \]

Common Occurrences of Lognormal Distribution in Nature

• Income distributions
• Stock prices
• Particle sizes
• Species abundance
• Chemical concentrations
• Length of sentences in writing

Uniform Distribution

• Simplest probability distribution
• Based on principle of indifference
• Fundamental in random number generation

Mathematical Formula

\[ f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \]

Common Occurrences of Uniform Distribution in Nature

• Random number generation
• Rounding errors
• Position of molecules in ideal gas
• Time of arrival within a fixed interval
• Quantum mechanical wavefunctions

Practical Examples: Normal Distribution

Practical Examples: Poisson Distribution

Statistical Tests and Applications

How do we choose the right distribution?

• Normal: Continuous, symmetric data
• Poisson: Count data, rare events
• Gamma: Positive, right-skewed data
• Lognormal: Multiplicative processes
• Uniform: Random selection, simulation

Important Considerations

• Sample size
• Data type
• Underlying process
• Assumptions validation

Real-World Examples

Random Variables in Daily Life

1. Weather Temperature
   • The daily high temperature in your city is a random variable
   • Each day’s temperature is an outcome that we can’t predict with certainty

plot(daily_temps, type = 'l', main = 'Daily Temperatures for a Month',
     xlab = 'Day', ylab = 'Temperature (°F)')

2. Customer Wait Times
   • The time customers wait in line at a coffee shop is a random variable
   • Each customer’s wait time is an outcome influenced by many random factors

Probability Distributions in Practice

1. Height Distribution in a Population
   • Heights in a population typically follow a normal (bell-shaped) distribution
   • This means most people are close to average height, with fewer people being very tall or very short

2. Number of Customers per Hour
   • The number of customers arriving at a store follows a Poisson distribution
   • This distribution is good for counting events that occur randomly over time

Estimation in Real Life

1. Estimating Average Monthly Spending
   • You want to know your average monthly spending
   • You look at your last 6 months of expenses to estimate it

## Monthly expenses: 1831.857 1930.947 2467.612 2021.153 2038.786 2514.519 \

## Estimated average monthly spending: 2134.146

2. Election Polling
   • Pollsters estimate the percentage of voters who support a candidate
   • They survey a sample of voters to estimate the true population proportion

## True support: 52 %\

## Poll estimate: 53.2 %

Key Takeaways

1. Random Variables
   • Are quantities that take random values
   • Examples: dice rolls, temperatures, wait times, heights
   • We can’t predict their exact values, but we can describe their patterns
2. Probability Distributions
   • Show how likely different values are
   • Different shapes for different types of data
   • Examples: normal (heights), Poisson (arrivals), exponential (wait times)
3. Estimation
   • Using sample data to guess population values
   • More data usually means better estimates
   • Always involves some uncertainty
   • Examples: polls, surveys, sample means