Discrete random variable distributions
Bernoulli trial
An experiment (one single trial) that can result in either a success (1) or a failure (0).
X ~ Bern(p) with p being the probability of success.
With replacement Without replacement
Fixed number of Binomial Hypergeometric
trials Multiple trials with the same Urn filled with white and black balls,
success probability with X the then drawing n balls out with X being
number of successes. the number of white balls from the
sample.
X ~ Bin(n, p)
n -> number of trials X ~ HGeom(w, b, n)
p -> probability of success w -> number of white balls
b -> number of black balls
n -> number of draws
Fixed number of Negative Binomial Negative Hypergeometric
successes Multiple trials with the same Urn filled with white and black balls,
success probability with X the then drawing n balls out until r white
number of failures before the rth balls have been obtained with X being
success. the number of black balls drawn before
drawing the rth white ball.
X ~ NBin(r, p)
r -> number of successes X ~ NHGeom(w, b, r)
p -> success probability w -> number of white balls
b -> number of black balls
r -> number of wanted white balls
Discrete Uniform
Picking a random number from some finite set of numbers with equal probability.
X is the chosen number.
X ~ DUnif(C) with C the finite, nonempty set of numbers.
First Success
A sequence of independent Bernoulli trials, each with the same success probability, with trials
until the first successful trial, including the success.
Y is the number of trials until the first trial, including the success.
Y ~ FS(p) with p the success probability.
Geometric
A sequence of independent Bernoulli trials, each with the same success probability, with trials
performed until a success occurs.
X is the number of failures before the first successful trial.
X ~ Geom(p) with p the success probability.
Bernoulli trial
An experiment (one single trial) that can result in either a success (1) or a failure (0).
X ~ Bern(p) with p being the probability of success.
With replacement Without replacement
Fixed number of Binomial Hypergeometric
trials Multiple trials with the same Urn filled with white and black balls,
success probability with X the then drawing n balls out with X being
number of successes. the number of white balls from the
sample.
X ~ Bin(n, p)
n -> number of trials X ~ HGeom(w, b, n)
p -> probability of success w -> number of white balls
b -> number of black balls
n -> number of draws
Fixed number of Negative Binomial Negative Hypergeometric
successes Multiple trials with the same Urn filled with white and black balls,
success probability with X the then drawing n balls out until r white
number of failures before the rth balls have been obtained with X being
success. the number of black balls drawn before
drawing the rth white ball.
X ~ NBin(r, p)
r -> number of successes X ~ NHGeom(w, b, r)
p -> success probability w -> number of white balls
b -> number of black balls
r -> number of wanted white balls
Discrete Uniform
Picking a random number from some finite set of numbers with equal probability.
X is the chosen number.
X ~ DUnif(C) with C the finite, nonempty set of numbers.
First Success
A sequence of independent Bernoulli trials, each with the same success probability, with trials
until the first successful trial, including the success.
Y is the number of trials until the first trial, including the success.
Y ~ FS(p) with p the success probability.
Geometric
A sequence of independent Bernoulli trials, each with the same success probability, with trials
performed until a success occurs.
X is the number of failures before the first successful trial.
X ~ Geom(p) with p the success probability.
