probability distribution Flashcards
notation, mean, and variance
What is a Discrete Uniform distribution?
Defined over a finite set of values where all outcomes are equally likely
Example: rolling a fair six-sided die.
Notation for a Discrete Uniform distribution?
X~D.Uniform(a,b)
a - first number in the range, b - ending element count.
What is a Bernoulli distribution?
Consists of 1 trial with two possible outcomes (success/failure)
Example: checking if a light bulb works.
Notation for a Bernoulli distribution?
X~Bernoulli(p)
p - probability of success.
What is a Binomial distribution?
Consists of n trials with two possible outcomes
Example: getting heads in 10 coin flips.
Notation for a Binomial distribution?
X~Binomial(n,p)
n - total number of trials, p - probability of success.
What is a Hypergeometric distribution?
Sampling individuals from a population without replacement
Example: selecting a group of organisms from a habitat.
Notation for a Hypergeometric distribution?
X~Hypergeom(D,N,n)
D - specific subset of N with the specific trait of interest, N - total number of elements in the population, n - sample size.
What is a Geometric distribution?
Counts the number of trials until the first success
Example: rolling a die until you get a 6.
Notation for a Geometric distribution?
X~Geom(p)
p - probability of success.
What is a Negative Binomial distribution?
Counts trials until reaching a certain number of successes
Example: rolling a die until you get three 6s.
Notation for a Negative Binomial distribution?
X~NB(r,p)
r - total number of successes, p - probability of success.
What is a Poisson distribution?
Counts how often something happens in a set amount of time or space
Example: number of emails received per hour.
Notation for a Poisson distribution?
X~Poisson(λ)
λ - Poisson mean or average number of successes.
What is a Multinomial distribution?
Similar to binomial but with more than two possible outcomes
Example: rolling a die multiple times and counting how often 1, 3, and 6 appears.
Notation for a Multinomial distribution?
X~Multinom(n,p1,p2,..,pk)
n - total number of trials, p1, p2, …, pk - probabilities of success for each outcome.
What is a Normal distribution?
Describes how many times a process repeats before obtaining a value within a specified range
Example: measuring heights of individuals.
Notation for a Normal distribution?
X~Normal(μ,σ^2)
μ - mean, σ^2 - variance
What is a Standard Normal distribution?
Normal distribution with a mean of 0 and a standard deviation of 1.
Notation for a Standard Normal distribution?
Z~Normal(0,1)
What is an Exponential distribution?
Counts the number of occurrences of an event within a fixed interval
Example: counting how many emissions occur in a 10-second window.
Notation for an Exponential distribution?
X~Exp(λ)
λ - Poisson mean or average number of successes in the specified interval.
Mean and Variance of Bernoulli
[𝑋] = 𝑝
𝑉𝑎𝑟[𝑋] = 𝑝(1 − 𝑝)
Mean and Variance of Binomial
𝐸[𝑋] = 𝑛𝑝
𝑉𝑎𝑟[𝑋] = 𝑛𝑝(1 − 𝑝)
