Probability exam 2

Hypergeometric distribution

finding probabilities of choosing k from a subset (2 groups)

Geometric Distribution

number of trials required to reach the first success of a bernoulli; sampling without replacement

Expectation of Geom

1/p

Variance of Geom

1 − p/(p^2)

Binomial Distribution

models the number of success in a fixed number of independent bernoulli trials; sampling with replacement

Expectation of Bin

np

Variance of Bin

np(1 − p)

Bernoulli Distribution

a single trial with the only 2 outcomes: success or failure; sampling with replacement

Expectation of Ber

p

Variance of Ber

p(1 − p)

Expectation (discrete)

∑︂ kP (X = k)

Expectation (continuous)

∫︂xf (x) dx

Variance (discrete)

∑︂(k − μ)^2*P (X = k)

Variance (continuous)

∫(x − μ)^2f (x) dx

Variance (in terms of expectation)

E[X^2] − (E[X])^2

Uniform Distribution

a continuous function bounded by a given interval [a,b]

Unif Density Function

1/b−a

CDF of unif

x-a/b-a

Expectation of unif

a + b/(2)

Variance of unif

(b − a)^2/12

Gaussian definition/density function

ϕ(x) = 1/√2π e^(−x2/2)