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Formula Reference

Random Variables & Distributions Formulas

Comprehensive collection of essential formulas for random variables, probability distributions, expectation, variance, and sampling distributions

7 Categories52 Formulas
Random Variable Fundamentals
Basic definitions and properties of random variables
4 formulas

Random Variable Definition

ξ:Ω→R,ξ−1(B)={ω:ξ(ω)∈B}∈F\xi: \Omega \to \mathbb{R}, \quad \xi^{-1}(B) = \{\omega : \xi(\omega) \in B\} \in \mathscr{F}

Maps sample space Ω to real numbers, measurable with respect to σ-field ℱ

Distribution Function (CDF)

F(x)=P(ξ≤x),x∈RF(x) = P(\xi \leq x), \quad x \in \mathbb{R}

Cumulative distribution function gives probability that random variable ≤ x

CDF Properties

Monotonic: F(a)≤F(b) if a≤bLimits: F(−∞)=0,F(+∞)=1Right continuous: F(x+)=F(x)\text{Monotonic: } F(a) \leq F(b) \text{ if } a \leq b \\ \text{Limits: } F(-\infty) = 0, F(+\infty) = 1 \\ \text{Right continuous: } F(x^+) = F(x)

Essential properties that any valid CDF must satisfy

Probability Intervals

P(a<ξ≤b)=F(b)−F(a)P(a < \xi \leq b) = F(b) - F(a)

Calculate probability of interval using CDF difference

Discrete Random Variables
Probability mass functions and discrete distributions
7 formulas

Probability Mass Function

P(ξ=xi)=p(xi),∑ip(xi)=1P(\xi = x_i) = p(x_i), \quad \sum_{i} p(x_i) = 1

PMF gives probability at each discrete value, must sum to 1

Discrete CDF

F(x)=∑xi≤xp(xi)F(x) = \sum_{x_i \leq x} p(x_i)

CDF is sum of PMF values up to x (step function)

Bernoulli Distribution

ξ∼Ber(p):P(ξ=k)=pk(1−p)1−k,k∈{0,1}\xi \sim \text{Ber}(p): \quad P(\xi = k) = p^k(1-p)^{1-k}, \quad k \in \{0,1\}

Single trial success/failure with success probability p

Binomial Distribution

ξ∼B(n,p):P(ξ=k)=(nk)pk(1−p)n−k\xi \sim B(n,p): \quad P(\xi = k) = \binom{n}{k} p^k (1-p)^{n-k}

Number of successes in n independent Bernoulli trials

Poisson Distribution

ξ∼P(λ):P(ξ=k)=λke−λk!,k=0,1,2,…\xi \sim P(\lambda): \quad P(\xi = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0,1,2,\ldots

Models rare events occurrence with rate parameter λ

Geometric Distribution

ξ∼Geo(p):P(ξ=k)=(1−p)k−1p,k=1,2,3,…\xi \sim \text{Geo}(p): \quad P(\xi = k) = (1-p)^{k-1}p, \quad k = 1,2,3,\ldots

Number of trials until first success, memoryless property

Hypergeometric Distribution

ξ∼H(n,M,N):P(ξ=k)=(Mk)(N−Mn−k)(Nn)\xi \sim H(n,M,N): \quad P(\xi = k) = \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}

Sampling without replacement: k successes in n draws from N items with M successes

Continuous Random Variables
Probability density functions and continuous distributions
9 formulas

Probability Density Function

p(x)≥0,∫−∞∞p(x)dx=1p(x) \geq 0, \quad \int_{-\infty}^{\infty} p(x) dx = 1

PDF is non-negative and integrates to 1, P(X=c) = 0 for continuous variables

Continuous CDF

F(x)=∫−∞xp(t)dt,p(x)=F′(x) (where F is differentiable)F(x) = \int_{-\infty}^x p(t) dt, \quad p(x) = F'(x) \text{ (where } F \text{ is differentiable)}

CDF is integral of PDF, PDF is derivative of CDF

Probability Intervals

P(a<ξ<b)=P(a≤ξ≤b)=∫abp(x)dx=F(b)−F(a)P(a < \xi < b) = P(a \leq \xi \leq b) = \int_a^b p(x) dx = F(b) - F(a)

Probability of interval equals area under PDF or CDF difference

Uniform Distribution

ξ∼U(a,b):p(x)={1b−aa<x<b0otherwise\xi \sim U(a,b): \quad p(x) = \begin{cases} \frac{1}{b-a} & a < x < b \\ 0 & \text{otherwise} \end{cases}

Constant probability density over interval [a,b]

Normal Distribution

ξ∼N(μ,σ2):p(x)=12πσe−(x−μ)22σ2\xi \sim N(\mu, \sigma^2): \quad p(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Bell-shaped distribution with mean μ and variance σ²

Standard Normal

Z∼N(0,1):p(z)=12πe−z22,Φ(z)=∫−∞zp(t)dtZ \sim N(0,1): \quad p(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}, \quad \Phi(z) = \int_{-\infty}^z p(t) dt

Standardized normal distribution, Φ(z) is standard normal CDF

Normal Standardization

If ξ∼N(μ,σ2), then Z=ξ−μσ∼N(0,1)\text{If } \xi \sim N(\mu, \sigma^2), \text{ then } Z = \frac{\xi - \mu}{\sigma} \sim N(0,1)

Transform any normal distribution to standard normal

Exponential Distribution

ξ∼Exp(λ):p(x)={λe−λxx>00x≤0\xi \sim \text{Exp}(\lambda): \quad p(x) = \begin{cases} \lambda e^{-\lambda x} & x > 0 \\ 0 & x \leq 0 \end{cases}

Models waiting times and lifetimes, memoryless property

Gamma Distribution

ξ∼Γ(α,β):p(x)=βαΓ(α)xα−1e−βx,x>0\xi \sim \Gamma(\alpha, \beta): \quad p(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0

Generalizes exponential distribution, α is shape, β is rate parameter

Expectation and Moments
Expected values, variance, and moment calculations
8 formulas

Discrete Expectation

E[ξ]=∑ixip(xi)E[\xi] = \sum_{i} x_i p(x_i)

Expected value for discrete random variable

Continuous Expectation

E[ξ]=∫−∞∞xp(x)dxE[\xi] = \int_{-\infty}^{\infty} x p(x) dx

Expected value for continuous random variable

Expectation of Function

E[g(ξ)]={∑ig(xi)p(xi)discrete∫−∞∞g(x)p(x)dxcontinuousE[g(\xi)] = \begin{cases} \sum_i g(x_i) p(x_i) & \text{discrete} \\ \int_{-\infty}^{\infty} g(x) p(x) dx & \text{continuous} \end{cases}

Expected value of function of random variable

Variance Definition

Var(ξ)=E[(ξ−E[ξ])2]=E[ξ2]−(E[ξ])2\text{Var}(\xi) = E[(\xi - E[\xi])^2] = E[\xi^2] - (E[\xi])^2

Variance measures spread around the mean

Standard Deviation

σ(ξ)=Var(ξ)\sigma(\xi) = \sqrt{\text{Var}(\xi)}

Standard deviation is square root of variance

Linearity of Expectation

E[aξ+bη+c]=aE[ξ]+bE[η]+cE[a\xi + b\eta + c] = aE[\xi] + bE[\eta] + c

Expectation is linear (holds regardless of independence)

Variance Properties

Var(aξ+b)=a2Var(ξ)Var(ξ+η)=Var(ξ)+Var(η)+2Cov(ξ,η)\text{Var}(a\xi + b) = a^2 \text{Var}(\xi) \\ \text{Var}(\xi + \eta) = \text{Var}(\xi) + \text{Var}(\eta) + 2\text{Cov}(\xi, \eta)

Variance of linear transformation and sum (covariance term for dependence)

k-th Moment

μk=E[ξk]={∑ixikp(xi)discrete∫−∞∞xkp(x)dxcontinuous\mu_k = E[\xi^k] = \begin{cases} \sum_i x_i^k p(x_i) & \text{discrete} \\ \int_{-\infty}^{\infty} x^k p(x) dx & \text{continuous} \end{cases}

k-th moment about origin

Joint Distributions
Multidimensional random variables and independence
8 formulas

Joint PMF (Discrete)

P(ξ=xi,η=yj)=pij,∑i∑jpij=1P(\xi = x_i, \eta = y_j) = p_{ij}, \quad \sum_i \sum_j p_{ij} = 1

Joint probability mass function for discrete random vector

Joint PDF (Continuous)

p(x,y)≥0,∫−∞∞∫−∞∞p(x,y)dxdy=1p(x,y) \geq 0, \quad \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x,y) dx dy = 1

Joint probability density function for continuous random vector

Marginal Distributions

Discrete: P(ξ=xi)=∑jpijContinuous: pξ(x)=∫−∞∞p(x,y)dy\text{Discrete: } P(\xi = x_i) = \sum_j p_{ij} \\ \text{Continuous: } p_\xi(x) = \int_{-\infty}^{\infty} p(x,y) dy

Marginal distributions obtained by summing/integrating joint distribution

Independence Condition

Discrete: pij=pipj for all i,jContinuous: p(x,y)=pξ(x)pη(y) a.e.\text{Discrete: } p_{ij} = p_i p_j \text{ for all } i,j \\ \text{Continuous: } p(x,y) = p_\xi(x) p_\eta(y) \text{ a.e.}

Random variables are independent if joint equals product of marginals

Conditional PMF

P(η=yj∣ξ=xi)=pijpi(pi>0)P(\eta = y_j | \xi = x_i) = \frac{p_{ij}}{p_i} \quad (p_i > 0)

Conditional probability mass function

Conditional PDF

pη∣ξ(y∣x)=p(x,y)pξ(x)(pξ(x)>0)p_{\eta|\xi}(y|x) = \frac{p(x,y)}{p_\xi(x)} \quad (p_\xi(x) > 0)

Conditional probability density function

Covariance

Cov(ξ,η)=E[(ξ−E[ξ])(η−E[η])]=E[ξη]−E[ξ]E[η]\text{Cov}(\xi, \eta) = E[(\xi - E[\xi])(\eta - E[\eta])] = E[\xi \eta] - E[\xi]E[\eta]

Measures linear dependence between random variables

Correlation Coefficient

ρ(ξ,η)=Cov(ξ,η)Var(ξ)Var(η),−1≤ρ≤1\rho(\xi, \eta) = \frac{\text{Cov}(\xi, \eta)}{\sqrt{\text{Var}(\xi)\text{Var}(\eta)}}, \quad -1 \leq \rho \leq 1

Standardized measure of linear dependence

Sampling Distributions
Chi-squared, t-distribution, and F-distribution
8 formulas

Chi-squared Distribution

If X1,…,Xn∼N(0,1) independent, then Y=∑i=1nXi2∼χ2(n)\text{If } X_1, \ldots, X_n \sim N(0,1) \text{ independent, then } Y = \sum_{i=1}^n X_i^2 \sim \chi^2(n)

Sum of squares of independent standard normal variables

Chi-squared PDF

χ2(n):p(x)=(1/2)n/2Γ(n/2)xn/2−1e−x/2,x>0\chi^2(n): \quad p(x) = \frac{(1/2)^{n/2}}{\Gamma(n/2)} x^{n/2-1} e^{-x/2}, \quad x > 0

PDF of chi-squared distribution with n degrees of freedom

Chi-squared Properties

E[χ2(n)]=n,Var[χ2(n)]=2nχ2(n1)+χ2(n2)=χ2(n1+n2) (independence)E[\chi^2(n)] = n, \quad \text{Var}[\chi^2(n)] = 2n \\ \chi^2(n_1) + \chi^2(n_2) = \chi^2(n_1 + n_2) \text{ (independence)}

Mean, variance, and additivity property of chi-squared

t-Distribution Definition

If X∼N(0,1),Y∼χ2(n) independent, then T=XY/n∼t(n)\text{If } X \sim N(0,1), Y \sim \chi^2(n) \text{ independent, then } T = \frac{X}{\sqrt{Y/n}} \sim t(n)

Ratio of standard normal to square root of scaled chi-squared

t-Distribution PDF

t(n):p(t)=Γ((n+1)/2)nπΓ(n/2)(1+t2n)−(n+1)/2t(n): \quad p(t) = \frac{\Gamma((n+1)/2)}{\sqrt{n\pi}\Gamma(n/2)} \left(1 + \frac{t^2}{n}\right)^{-(n+1)/2}

PDF of t-distribution, approaches standard normal as n → ∞

F-Distribution Definition

If X∼χ2(m),Y∼χ2(n) independent, then F=X/mY/n∼F(m,n)\text{If } X \sim \chi^2(m), Y \sim \chi^2(n) \text{ independent, then } F = \frac{X/m}{Y/n} \sim F(m,n)

Ratio of two independent scaled chi-squared variables

F-Distribution PDF

F(m,n):p(f)=Γ((m+n)/2)Γ(m/2)Γ(n/2)mm/2nn/2(mf+n)(m+n)/2fm/2−1F(m,n): \quad p(f) = \frac{\Gamma((m+n)/2)}{\Gamma(m/2)\Gamma(n/2)} \frac{m^{m/2} n^{n/2}}{(mf+n)^{(m+n)/2}} f^{m/2-1}

PDF of F-distribution with m and n degrees of freedom

Distribution Relationships

If T∼t(n), then T2∼F(1,n)If F∼F(m,n), then 1/F∼F(n,m)\text{If } T \sim t(n), \text{ then } T^2 \sim F(1,n) \\ \text{If } F \sim F(m,n), \text{ then } 1/F \sim F(n,m)

Key relationships between t and F distributions

Common Distribution Parameters
Mean and variance formulas for standard distributions
8 formulas

Bernoulli Parameters

ξ∼Ber(p):E[ξ]=p,Var(ξ)=p(1−p)\xi \sim \text{Ber}(p): \quad E[\xi] = p, \quad \text{Var}(\xi) = p(1-p)

Mean and variance for Bernoulli distribution

Binomial Parameters

ξ∼B(n,p):E[ξ]=np,Var(ξ)=np(1−p)\xi \sim B(n,p): \quad E[\xi] = np, \quad \text{Var}(\xi) = np(1-p)

Mean and variance for binomial distribution

Poisson Parameters

ξ∼P(λ):E[ξ]=λ,Var(ξ)=λ\xi \sim P(\lambda): \quad E[\xi] = \lambda, \quad \text{Var}(\xi) = \lambda

Mean equals variance for Poisson distribution

Geometric Parameters

ξ∼Geo(p):E[ξ]=1p,Var(ξ)=1−pp2\xi \sim \text{Geo}(p): \quad E[\xi] = \frac{1}{p}, \quad \text{Var}(\xi) = \frac{1-p}{p^2}

Mean and variance for geometric distribution

Uniform Parameters

ξ∼U(a,b):E[ξ]=a+b2,Var(ξ)=(b−a)212\xi \sim U(a,b): \quad E[\xi] = \frac{a+b}{2}, \quad \text{Var}(\xi) = \frac{(b-a)^2}{12}

Mean and variance for uniform distribution

Normal Parameters

ξ∼N(μ,σ2):E[ξ]=μ,Var(ξ)=σ2\xi \sim N(\mu, \sigma^2): \quad E[\xi] = \mu, \quad \text{Var}(\xi) = \sigma^2

Distribution parameters directly give mean and variance

Exponential Parameters

ξ∼Exp(λ):E[ξ]=1λ,Var(ξ)=1λ2\xi \sim \text{Exp}(\lambda): \quad E[\xi] = \frac{1}{\lambda}, \quad \text{Var}(\xi) = \frac{1}{\lambda^2}

Mean and variance for exponential distribution

Gamma Parameters

ξ∼Γ(α,β):E[ξ]=αβ,Var(ξ)=αβ2\xi \sim \Gamma(\alpha, \beta): \quad E[\xi] = \frac{\alpha}{\beta}, \quad \text{Var}(\xi) = \frac{\alpha}{\beta^2}

Mean and variance for gamma distribution

📐 Quick Reference Guide

Essential tips for working with random variables and probability distributions

Distribution Identification

  • • Discrete: Countable outcomes → Use PMF
  • • Continuous: Interval values → Use PDF
  • • Normal: Bell-shaped, symmetric data
  • • Exponential: Waiting times, memoryless
  • • Binomial: Fixed trials, success/failure
  • • Poisson: Rare events, fixed rate

Key Relationships

  • • Independence: Joint = Product of marginals
  • • Standardization: (X-μ)/σ ~ N(0,1)
  • • Variance: Var(X) = E[X²] - (E[X])²
  • • Linear transform: E[aX+b] = aE[X]+b
  • • Chi-squared: Sum of squared normals
  • • t-distribution: Small sample inference

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