Random Variables & Distributions Formulas

Random Variable Fundamentals

Basic definitions and properties of random variables

4 formulas

Random Variable Definition

\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(\xi \leq x), \quad x \in \mathbb{R}

$Cumulative distribution function gives probability that random variable ≤ x$

CDF Properties

\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 < \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(\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) = \sum_{x_i \leq x} p(x_i)

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

Bernoulli Distribution

\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

\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

\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

\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

\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) \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) = \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 < \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

\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

\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 \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

\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

\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

\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[\xi] = \sum_{i} x_i p(x_i)

$Expected value for discrete random variable$

Continuous Expectation

E[\xi] = \int_{-\infty}^{\infty} x p(x) dx

$Expected value for continuous random variable$

Expectation of Function

E[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

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

$Variance measures spread around the mean$

Standard Deviation

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

$Standard deviation is square root of variance$

Linearity of Expectation

E[a\xi + b\eta + c] = aE[\xi] + bE[\eta] + c

$Expectation is linear (holds regardless of independence)$

Variance Properties

\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

\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(\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) \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

\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

\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(\eta = y_j | \xi = x_i) = \frac{p_{ij}}{p_i} \quad (p_i > 0)

$Conditional probability mass function$

Conditional PDF

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

$Conditional probability density function$

Covariance

\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

\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

\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

\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[\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

\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): \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

\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): \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

\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

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

$Mean and variance for Bernoulli distribution$

Binomial Parameters

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

$Mean and variance for binomial distribution$

Poisson Parameters

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

$Mean equals variance for Poisson distribution$

Geometric Parameters

\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

\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

\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

\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

\xi \sim \Gamma(\alpha, \beta): \quad E[\xi] = \frac{\alpha}{\beta}, \quad \text{Var}(\xi) = \frac{\alpha}{\beta^2}

$Mean and variance for gamma distribution$

Random Variable Definition

Distribution Function (CDF)

CDF Properties

Probability Intervals

Probability Mass Function

Discrete CDF

Bernoulli Distribution

Binomial Distribution

Poisson Distribution

Geometric Distribution

Hypergeometric Distribution

Probability Density Function

Continuous CDF

Probability Intervals

Uniform Distribution

Normal Distribution

Standard Normal

Normal Standardization

Exponential Distribution

Gamma Distribution

Discrete Expectation

Continuous Expectation

Expectation of Function

Variance Definition

Standard Deviation

Linearity of Expectation

Variance Properties

k-th Moment

Joint PMF (Discrete)

Joint PDF (Continuous)

Marginal Distributions

Independence Condition

Conditional PMF

Conditional PDF

Covariance

Correlation Coefficient

Chi-squared Distribution

Chi-squared PDF

Chi-squared Properties

t-Distribution Definition

t-Distribution PDF

F-Distribution Definition

F-Distribution PDF

Distribution Relationships

Bernoulli Parameters

Binomial Parameters

Poisson Parameters

Geometric Parameters

Uniform Parameters

Normal Parameters

Exponential Parameters

Gamma Parameters

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