Named Distributions Flashcards
(25 cards)
Mean of Bern(p)
p
Mean of Binomial(n, p)
np
Mean of Geometric(p)
1/p
Mean of Poisson(λ)
λ
Mean of Expo(λ)
1/λ
Mean of Normal(μ, σ^2)
μ
Mean of Uniform(a,b)
(a+b)/2
Example of Mean of DUniform(1,2,3,4,5,6)
(1 + 2+ 3 + 4 + 5 + 6)/6
PMF of DUnif(C)
1/|C|(|C| is the length of C)
PDF of Unif(a,b)
1/(b-a), x ∈ (a, b)
Story of Binomial
n independent Bernoulli trials
Story of Hypergeometric, X ∼ HGeom(w, b, n).
Consider an urn with w white balls
and b black balls. We draw n balls out of the urn at random without replacement. X is the number of white balls in the sample
Story of Geometric
Consider a sequence of independent
Bernoulli trials, each with the same success probability p ∈ (0, 1), with trials per-
formed until a success occurs. Let X be the number of failures before the first
successful trial.
Story of Negative Binomial
In a sequence of independent
Bernoulli trials with success probability p, if X is the number of failures before
the rth success, then X is said to have the Negative Binomial distribution with
parameters r and p, denoted X ∼ NBin(r, p).
Story of Poisson
Probability of a given number of events occuring in a fixed time interval, with the expected number of events being λ
Story of Exponential
Analogous to Geometric, but we are now waiting for a success in continuous
time, where successes arrive at a rate of λ successes per unit of time
Variance of Bernoulli
pq
Variance of Binomial
npq
Variance of Geometric
q/p^2
Variance of Poisson
λ
Variance of Exponential
1/λ^2
Variance of Normal
σ^2
Convolution sums
Let X and Y be independent
r.v.s and T = X + Y be their sum. If X and Y are discrete, then the PMF of T is
P (T = t) = (for all x)∑(Y = t − x)P (X = x)
= (for all y)∑P (X = t − y)P (Y = y).
Convolution integral
f(t) =∫fY(t-x)fX(x)dx = ∫fX(t-y)fX(y)dy
