Chapter 5: Useful Probability Distributions#
Most random phenomena we encounter fall into a small number of standard patterns. Rather than deriving probabilities from first principles every time, we can use standard probability distributions—well-studied functions that model common situations.
Knowing these distributions is like having a toolkit:
Recognize patterns in real problems
Use established formulas instead of deriving from scratch
Leverage existing theory and computational tools
Communicate efficiently with other practitioners
Why Study Probability Distributions?#
Consider these scenarios:
Number of heads in 100 coin flips
Time until the next customer arrives
Number of typos on a page
Height of randomly selected person
Number of failures before first success
Each of these follows a named probability distribution with known properties. Once you recognize the pattern, you immediately know:
The probability mass/density function
The expected value and variance
How to generate samples
What approximations are available
What You’ll Learn#
In this chapter, you will learn:
Discrete Distributions (Section 5.1)
Discrete Uniform
Bernoulli
Geometric
Binomial
Multinomial
Poisson
Continuous Distributions (Section 5.2)
Continuous Uniform
Beta
Gamma
Exponential
The Normal Distribution (Section 5.3)
Standard Normal
General Normal
Properties and applications
The 68-95-99.7 rule
Normal Approximation to Binomial (Section 5.4)
When to use the approximation
How to apply it correctly
Continuity correction
Chapter Structure#
Common Distributions Overview#
Distribution |
Type |
Parameters |
Used For |
|---|---|---|---|
Uniform |
Discrete |
\(n\) |
Equal probabilities |
Bernoulli |
Discrete |
\(p\) |
Single trial (success/fail) |
Binomial |
Discrete |
\(n, p\) |
Number of successes in \(n\) trials |
Geometric |
Discrete |
\(p\) |
Trials until first success |
Poisson |
Discrete |
\(\lambda\) |
Count of rare events |
Uniform |
Continuous |
\(a, b\) |
Equal density over interval |
Exponential |
Continuous |
\(\lambda\) |
Time until event |
Normal |
Continuous |
\(\mu, \sigma^2\) |
Natural variation, sums |
Key Insight#
{admonition} Pattern Recognition :class: tip Most of statistics and probability involves:
Recognizing which distribution fits your problem
Identifying the parameters from context
Applying formulas for that distribution
Using computational tools for calculations
You don’t memorize—you recognize and look up!
Prerequisites#
Before starting this chapter, you should understand:
Random variables (Chapter 4.1)
Expected values and variance (Chapter 4.2)
Discrete vs. continuous probability
Let’s begin by exploring discrete probability distributions!