Discrete Probability Distributions (Probability Mass Functions)

used in scenarios where the set of possible outcomes are discrete, either:
- a finite number of values (e.g. coin toss or dice roll)
- an infinite sequence of values (e.g. counting numbers)
encoding a discrete list of the probabilities of the outcomes

Discrete Probability Distributions - Calculating Statistics

see: Discrete Probability Distribution - Calculating Statistics

Discrete Probability Distributions - Types

Discrete Distributions	# of Possible Outcomes Per Trial	# of Trails	# of Success	Description
Bernoulli Distribution	=2	=1	Random	simplest discrete probability distribution
Binomial Distribution	=2	≥1	Random	number of successes in 𝑛 trials number of failures in 𝑛 trails a Binomial variable = sum of independent Bernoulli variables
Geometric Distribution	=2	Random	=1	number of failures before the first success number of successes before the first failure
Negative Binomial (Pascal) Distribution	=2	Random	≥1	number of failures before the 𝑥^𝑡𝘩 success number of successes before the 𝑥^𝑡𝘩 failure a generalization of the geometric distribution to cases where the number of successes ≥1 instead of 1 a Negative Binomial variable = sum of independent Geometric variables
Multinoulli Distribution	≥1	=1	Random	a generalization of the bernoulli distribution to cases where the discrete random variable has ≥1 possible outcomes per trial instead of 2
Multinomial Distribution	≥1	≥1	Random	a generalization of the binomial distribution to cases where the discrete random variable has ≥1 possible outcomes per trial instead of 2
Negative Multinomial Distribution	≥1	Random	≥1	a generalization of the negative binomial distribution to cases where the discrete random variable has ≥1 possible outcomes per trial instead of 2
Discrete Uniform Distribution	≥1	=1	Random	a specific case of Multinoulli distribution where every outcome has an equal probability

Discrete Probability Distributions - Other Types

Discrete Distributions

# of Events

(discrete)

Time Interval

(continuous)

Description

Poisson Distribution

Random

Fixed

given a mean number of events that happen within a time period, the number of events occurring within a time period has Poisson Distribution
the time between events has an Exponential Distribution

Hypergeometric Distribution

used to calculate probabilities of combination without replacement

Multivariate Hypergeometric Distribution

a generalization of the hypergeometric distribution