STAT 101
M04 · L02
Module 4 · Lesson 2

Common Discrete Distributions

Five families. Five stories. Each encodes a different way that counting experiments generate data.

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STAT 101
M04 · L02
Why named distributions?

Each Distribution Tells a Story

Recognize the story behind your data and you immediately know the model, the mean, the variance, and how to compute any probability.

The five stories
Bernoulli → Binomial → Poisson
Geometric → Negative Binomial
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STAT 101
M04 · L02
The atom of all discrete distributions

Bernoulli: One Trial

X = 1 (success) with probability p, X = 0 (failure) with probability 1 − p.

Mean
E[X] = p
Variance
p(1 − p)
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STAT 101
M04 · L02
n independent Bernoulli trials

Binomial: Count of Successes

Repeat a Bernoulli trial n times. Count total successes. The most important discrete distribution in statistics.

PMF
P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}
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STAT 101
M04 · L02
Binomial properties

Binomial: Mean & Variance

Mean
E[X] = np
Variance
np(1−p)
Example
200 emails, p = 0.30
E[purchases] = 60, σ ≈ 6.5
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STAT 101
M04 · L02
Rare, independent events in a fixed interval

Poisson: Rare Events

Phone calls per hour. Accidents per month. Mutations per genome. One parameter λ controls everything.

  • Events occur independently
  • Constant rate λ over the interval
  • No simultaneous events
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STAT 101
M04 · L02
A unique property

Poisson: Mean = Variance

PMF
P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}
Mean
E[X] = λ
Variance
Var(X) = λ
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STAT 101
M04 · L02
How long until success?

Geometric: First Success

X = number of trials until the first success. The unique memoryless discrete distribution.

Mean
E[X] = 1/p
Variance
(1−p)/p²
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STAT 101
M04 · L02
Generalization of Geometric

Negative Binomial: r-th Success

X = trials until the r-th success. Sum of r independent Geometric(p) variables.

Mean
E[X] = r/p
Variance
r(1−p)/p²
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STAT 101
M04 · L02
Pick the right model

Choosing the Right Distribution

  • Single binary trial → Bernoulli
  • Fixed n, count successes → Binomial
  • Rare events in interval → Poisson
  • Trials to 1st success → Geometric
  • Trials to r-th success → Neg. Binomial
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STAT 101
M04 · L02
A quick diagnostic

Mean vs. Variance

  • Mean ≈ Variance ⇒ Poisson
  • Variance ≫ Mean ⇒ Negative Binomial
  • Fixed n, count ⇒ Binomial
  • Waiting time ⇒ Geometric / Neg. Binomial
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STAT 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

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STAT 101
Summary
Recap

What you learned

Bernoulli, Binomial, Poisson, Geometric, Negative Binomial — five distributions, five stories. Each tells you how counting experiments work.

Up next
L3: Common Continuous Distributions →
All lessons Read article
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