Probability Distributions — Formula Sheet

Probability Distributions — Formula Sheet

Random variable basics

\[ X:S\to\mathbb{R}, \qquad \text{discrete: countable values}, \qquad \text{continuous: } P(X=a)=0. \]

Discrete: pmf and cdf

\[ f(x_k)=P(X=x_k)\ge 0, \qquad \sum_k f(x_k)=1. \] \[ F(x)=P(X\le x)=\sum_{x_k\le x} f(x_k), \qquad f(x_i)=F(x_i)-F(x_{i-1}). \]

Continuous: pdf and cdf

\[ f(x)\ge 0, \qquad \int_{-\infty}^{\infty} f(x)\,dx=1, \qquad P(a\le X\le b)=\int_a^b f(x)\,dx. \] \[ F(x)=\int_{-\infty}^{x} f(u)\,du, \qquad f(x)=\dfrac{dF(x)}{dx}=F'(x). \]

Expectation and variance

\[ E(X)=\sum_k x_k f(x_k) \quad\text{or}\quad \int_{-\infty}^{\infty} x\,f(x)\,dx. \] \[ E\big(g(X)\big)=\sum_k g(x_k) f(x_k) \quad\text{or}\quad \int_{-\infty}^{\infty} g(x)\,f(x)\,dx. \] \[ \operatorname{Var}(X)=E\big[(X-\mu)^2\big]=E(X^2)-\big(E(X)\big)^2, \qquad \sigma=\sqrt{\operatorname{Var}(X)}. \]

Linear-transformation rules

\[ E(aX+b)=aE(X)+b, \qquad E(b)=b. \] \[ \operatorname{Var}(aX+b)=a^2\operatorname{Var}(X), \qquad \operatorname{Var}(b)=0. \]

\(k\)-th moment about the origin

\[ \mu_k'=E(X^k). \]

One-point distribution

\[ P(X=x_0)=1, \qquad \mu=x_0, \qquad \sigma^2=0. \]

Two-point distribution

\[ P(X=x_1)=p, \quad P(X=x_2)=q=1-p, \qquad \mu=px_1+qx_2. \]

Bernoulli distribution \(\text{Ber}(p)\)

\[ f(x)=p^x(1-p)^{1-x}, \quad x\in\{0,1\}, \qquad \mu=p, \quad \sigma^2=pq. \]

Binomial distribution \(B(n,p)\)

\[ P(X=x)=\binom{n}{x}p^x q^{\,n-x}, \quad x=0,1,2,\dots,n. \] \[ \mu=np, \qquad \sigma^2=npq, \qquad \sigma=\sqrt{npq}. \] \[ P(X\ge 1)=1-P(X=0)=1-q^{\,n}. \]