Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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