Download Math 411, Continuous and discrete distributions of random variables.

Math 411, Continuous and discrete distributions of random variables.∗
Discrete distribution
P
pi = P (X = xi ), pi ≥ 0, pi = 1.
Distribution
ContinuousZdistribution
∞
f (x) dx = 1.
f (x), f (x) ≥ 0,
−∞
PDF
b
Z
probability density function
f (x) dx = F (b) − F (a).
P (a ≤ X ≤ b) =
a
Z
F (x) = P (−∞ < X ≤ x)
x
F (x) = P (−∞ < X ≤ x) =
f (y) dy,
−∞
=
X
P (X = xi )
CDF
cumulative distribution function
F 0 (x) = f (x)
xi ≤x
EX =
X
∞
Z
xi P (X = xi ).
Expected value
xf (x) dx.
EX =
−∞
Er(X) =
X
∞
Z
r(xi )P (X = xi ).
Er(X) =
r(x)f (x) dx.
−∞
ZZ
pij = P (X = xi , Y = yj )
Joint distribution
P ((X, Y ) ∈ A) =
f (x, y) dx dy.
A
P (X = x) =
X
Z
P (X = x, Y = y)
Marginal distribution
P (X = x, Y = y) = P (X = x)P (Y = y)
∗
f (x, y) dy.
−∞
y
P (X = x|Y = y) =
∞
fX (x) =
P (X = x, Y = y)
P (Y = y)
Independence
Conditional distribution
c 2015 by Michael Anshelevich.
1
f (x, y) = fX (x)fY (y).
fX (x|Y = y) =
f (x, y)
.
fY (y)
Some discrete distributions.
Binomial(n, p), n ≥ 1, 0 ≤ p ≤ 1. The number of successes in n trials, with a success having probability p.
P (X = k) = Cn,k pk (1 − p)n−k ,
0 ≤ k ≤ n,
EX = np,
Var [X] = np(1 − p).
Bernoulli(p) is Binomial(1, p). The number of successes in one trial, with a success having probability p.
P (X = 0) = 1 − p,
P (X = 1) = p,
EX = p,
Var [X] = p(1 − p).
Geometric(p), 0 ≤ p ≤ 1. The number of trials until a success occurs, with a success having probability p.
1
1−p
P (X = k) = (1 − p)k−1 p, k ≥ 1, EX = , Var [X] =
.
p
p2
Poisson(λ), λ > 0. The number of rare events in a given time period, with the average rate λ of the events.
λk
, k ≥ 0, EX = λ, Var [X] = λ.
k!
Negative binomial(n, p), n ≥ 1, 0 ≤ p ≤ 1. The number of trials until nth success occurs, with a success having
probability p. Geometric(p) is Negative binomial(1, p)
1−p
n
P (X = k) = Ck−1,n−1 (1 − p)k−n pn , k ≥ n, EX = , Var [X] = n 2 .
p
p
P (X = k) = e−λ
Some continuous distributions.
In all cases, “and zero otherwise” is omitted.
Uniform on [a, b]. A number picked at random between a and b.
1
a+b
(b − a)2
, a ≤ x ≤ b, EX =
, Var [X] =
.
b−a
2
12
Standard uniform is on [0, 1]. A number picked at random between 0 and 1.
1
1
f (x) = 1, 0 ≤ x ≤ 1, EX = , Var [X] = .
2
12
Gamma(n, λ), n ≥ 1, λ > 0. Waiting time until the n’th event, assuming the distribution is memoryless.
λn
n
n
f (x) =
xn−1 e−λx , x ≥ 0, EX = , Var [X] = 2 .
(n − 1)!
λ
λ
f (x) =
Exponential(λ) is Gamma(1, λ). Waiting time until the next event, assuming the distribution is memoryless.
1
1
f (x) = λe−λx , x ≥ 0, EX = , Var [X] = 2 .
λ
λ
Power laws with parameter (ρ), ρ > 1.
f (x) = (ρ − 1)x−ρ , x ≥ 1.
Normal distribution N (µ, σ 2 ).
1
2
2
f (x) = √
e−(x−µ) /2σ , −∞ < x < ∞, EX = µ, Var [X] = σ 2 .
2
2πσ
Standard normal is N (0, 1). Appears in the Central Limit Theorem.
1
2
f (x) = √ e−x /2 , −∞ < x < ∞, EX = 0, Var [X] = 1.
2π
Cauchy distribution.
1 1
f (x) =
, −∞ < x < ∞, EX undefined.
π 1 + x2