Digital Characteristics Formulas

Mathematical Expectation

Fundamental formulas for mathematical expectation and its properties

6 formulas

Discrete Expectation

E[\xi] = \sum_{k=1}^{\infty} x_k p(x_k)

$Expected value for discrete random variable with distribution P(ξ = xₖ) = p(xₖ)$

Continuous Expectation

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

$Expected value for continuous random variable with probability density function p(x)$

General Form (Stieltjes Integral)

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

$Unified form for both discrete and continuous random variables using distribution function F(x)$

Expectation of Function

E[g(\xi)] = \begin{cases} \sum_k g(x_k) p(x_k) & \text{discrete} \\ \int_{-\infty}^{\infty} g(x) p(x) dx & \text{continuous} \end{cases}

$Expected value of function g(ξ) of random variable ξ$

Linearity of Expectation

E\left[\sum_{i=1}^n c_i \xi_i + b\right] = \sum_{i=1}^n c_i E[\xi_i] + b

$Linear combinations preserve expectation (independence not required)$

Product of Independent Variables

\text{If } \xi_1, \ldots, \xi_n \text{ independent, then } E[\xi_1 \cdots \xi_n] = E[\xi_1] \cdots E[\xi_n]

$Expectation of product equals product of expectations for independent variables$

Variance and Standard Deviation

Variance formulas and properties for measuring dispersion

6 formulas

Variance Definition

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

$Variance as expectation of squared deviation from mean$

Computational Formula

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

$More convenient formula for calculating variance$

Standard Deviation

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

$Square root of variance, having same units as the random variable$

Variance of Linear Transformation

\text{Var}(a\xi + b) = a^2 \text{Var}(\xi)

$Translation doesn't affect variance, scaling affects quadratically$

Variance of Sum

\text{Var}\left(\sum_{i=1}^n \xi_i\right) = \sum_{i=1}^n \text{Var}(\xi_i) + 2\sum_{1 \leq i < j \leq n} \text{Cov}(\xi_i, \xi_j)

$General formula for variance of sum including covariance terms$

Variance of Independent Sum

\text{If } \xi_1, \ldots, \xi_n \text{ independent, then } \text{Var}\left(\sum_{i=1}^n \xi_i\right) = \sum_{i=1}^n \text{Var}(\xi_i)

$Variance is additive for independent random variables$

Covariance and Correlation

Measures of joint variability and linear dependence between variables

6 formulas

Covariance Definition

\text{Cov}(\xi, \eta) = E[(\xi - E[\xi])(\eta - E[\eta])]

$Measures how two variables jointly deviate from their respective means$

Computational Covariance

\text{Cov}(\xi, \eta) = E[\xi\eta] - E[\xi]E[\eta]

$More convenient formula for calculating covariance$

Correlation Coefficient

\rho(\xi, \eta) = \frac{\text{Cov}(\xi, \eta)}{\sqrt{\text{Var}(\xi) \cdot \text{Var}(\eta)}}

$Standardized measure of linear dependence, always between -1 and 1$

Covariance Properties

\text{Cov}(a\xi + b, c\eta + d) = ac \cdot \text{Cov}(\xi, \eta)

$Bilinearity of covariance with respect to linear transformations$

Bilinearity of Covariance

\text{Cov}\left(\sum_{i=1}^m \xi_i, \sum_{j=1}^n \eta_j\right) = \sum_{i=1}^m \sum_{j=1}^n \text{Cov}(\xi_i, \eta_j)

$Covariance distributes over sums$

Independence Condition

\xi \text{ and } \eta \text{ independent} \Rightarrow \text{Cov}(\xi, \eta) = 0

$Independent variables are uncorrelated (converse not generally true)$

Important Inequalities

Fundamental probability inequalities involving expectations and variances

5 formulas

Markov Inequality

P(|\xi| \geq \varepsilon) \leq \frac{E[|\xi|]}{\varepsilon}, \quad \varepsilon > 0

$Upper bound for tail probability using first moment$

Chebyshev Inequality

P(|\xi - E[\xi]| \geq \varepsilon) \leq \frac{\text{Var}(\xi)}{\varepsilon^2}, \quad \varepsilon > 0

$Probability bound for deviation from mean using variance$

Cauchy-Schwarz Inequality

|E[\xi\eta]|^2 \leq E[\xi^2] \cdot E[\eta^2]

$Fundamental inequality relating product expectation to second moments$

Jensen Inequality

\text{If } g(x) \text{ is convex, then } g(E[\xi]) \leq E[g(\xi)]

$Relates function of expectation to expectation of function for convex functions$

Hölder Inequality

E[|\xi\eta|] \leq (E[|\xi|^p])^{1/p} (E[|\eta|^q])^{1/q}, \quad \frac{1}{p} + \frac{1}{q} = 1

$Generalization of Cauchy-Schwarz inequality for higher moments$

Moments and Generating Functions

Moment formulas and generating function properties

7 formulas

Raw Moments

m_k = E[\xi^k], \quad k = 1, 2, 3, \ldots

$k-th moment about origin$

Central Moments

c_k = E[(\xi - E[\xi])^k], \quad k = 1, 2, 3, \ldots

$k-th moment about the mean$

Moment Relationships

c_1 = 0, \quad c_2 = \text{Var}(\xi), \quad m_1 = E[\xi]

$Special cases of first and second moments$

Skewness Coefficient

\gamma_1 = \frac{c_3}{c_2^{3/2}} = \frac{E[(\xi - E[\xi])^3]}{(\text{Var}(\xi))^{3/2}}

$Measures asymmetry of distribution (γ₁ > 0: right-skewed, γ₁ < 0: left-skewed)$

Kurtosis Coefficient

\gamma_2 = \frac{c_4}{c_2^2} - 3 = \frac{E[(\xi - E[\xi])^4]}{(\text{Var}(\xi))^2} - 3

$Measures peakedness relative to normal distribution (γ₂ > 0: leptokurtic, γ₂ < 0: platykurtic)$

Moment Generating Function

M_\xi(t) = E[e^{t\xi}], \quad \text{if exists}

$Generates all moments through derivatives at origin$

MGF Moment Formula

\text{If } M_\xi(t) \text{ exists, then } E[\xi^k] = M_\xi^{(k)}(0)

$k-th moment equals k-th derivative of MGF evaluated at zero$

Characteristic Functions

Characteristic function definitions, properties, and applications

8 formulas

Characteristic Function Definition

f_\xi(t) = E[e^{it\xi}] = \int_{-\infty}^{\infty} e^{itx} dF(x), \quad t \in \mathbb{R}

$Fourier transform of random variable, always exists$

Discrete Characteristic Function

f_\xi(t) = \sum_{k=1}^{\infty} e^{itx_k} p(x_k)

$Characteristic function for discrete random variables$

Continuous Characteristic Function

f_\xi(t) = \int_{-\infty}^{\infty} e^{itx} p(x) dx

$Characteristic function for continuous random variables$

Linear Transformation

\text{If } \eta = a\xi + b, \text{ then } f_\eta(t) = e^{itb} f_\xi(at)

$Characteristic function under linear transformation$

Independence Property

\text{If } \xi_1, \ldots, \xi_n \text{ independent and } \eta = \sum_{i=1}^n \xi_i, \text{ then } f_\eta(t) = \prod_{i=1}^n f_{\xi_i}(t)

$CF of sum equals product of CFs for independent variables$

Moment Generation from CF

\text{If } E[|\xi|^n] < \infty, \text{ then } E[\xi^k] = \frac{f_\xi^{(k)}(0)}{i^k}, \quad 0 \leq k \leq n

$Extract moments through derivatives of characteristic function at origin$

Inversion Formula

F(x_2) - F(x_1) = \lim_{T \to \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-itx_1} - e^{-itx_2}}{it} f_\xi(t) dt

$Recover distribution function from characteristic function$

Fourier Inversion

\text{If } \int_{-\infty}^{\infty} |f_\xi(t)| dt < \infty, \text{ then } p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-itx} f_\xi(t) dt

$Direct recovery of density from characteristic function$

Common Distribution Parameters

Expectation and variance formulas for standard probability distributions

8 formulas

Bernoulli Distribution Ber(p)

E[\xi] = p, \quad \text{Var}(\xi) = p(1-p), \quad f(t) = pe^{it} + (1-p)

$Single trial success/failure with success probability p$

Binomial Distribution B(n,p)

E[\xi] = np, \quad \text{Var}(\xi) = np(1-p), \quad f(t) = (pe^{it} + (1-p))^n

$Number of successes in n independent Bernoulli trials$

Poisson Distribution P(λ)

E[\xi] = \lambda, \quad \text{Var}(\xi) = \lambda, \quad f(t) = e^{\lambda(e^{it} - 1)}

$Rare events with rate parameter λ (mean equals variance)$

Geometric Distribution Geo(p)

E[\xi] = \frac{1}{p}, \quad \text{Var}(\xi) = \frac{1-p}{p^2}, \quad f(t) = \frac{pe^{it}}{1-(1-p)e^{it}}

$Number of trials until first success$

Uniform Distribution U[a,b]

E[\xi] = \frac{a+b}{2}, \quad \text{Var}(\xi) = \frac{(b-a)^2}{12}, \quad f(t) = \frac{e^{itb} - e^{ita}}{it(b-a)}

$Equal probability over interval [a,b]$

Exponential Distribution Exp(λ)

E[\xi] = \frac{1}{\lambda}, \quad \text{Var}(\xi) = \frac{1}{\lambda^2}, \quad f(t) = \frac{1}{1 - \frac{it}{\lambda}}

$Continuous analog of geometric distribution (memoryless property)$

Normal Distribution N(μ,σ²)

E[\xi] = \mu, \quad \text{Var}(\xi) = \sigma^2, \quad f(t) = e^{it\mu - \frac{\sigma^2 t^2}{2}}

$Bell-shaped distribution with location parameter μ and scale parameter σ$

Gamma Distribution Γ(α,β)

E[\xi] = \frac{\alpha}{\beta}, \quad \text{Var}(\xi) = \frac{\alpha}{\beta^2}, \quad f(t) = \left(1 - \frac{it}{\beta}\right)^{-\alpha}

$Generalizes exponential distribution with shape parameter α and rate parameter β$

Multivariate Extensions

Extensions to multivariate case and joint characteristics

6 formulas

Joint Expectation

E[g(\xi, \eta)] = \begin{cases} \sum_i \sum_j g(x_i, y_j) p_{ij} & \text{discrete} \\ \int\int g(x,y) p(x,y) dx dy & \text{continuous} \end{cases}

$Expected value of function of two random variables$

Covariance Matrix

\boldsymbol{\Sigma} = E[(\boldsymbol{\xi} - E[\boldsymbol{\xi}])(\boldsymbol{\xi} - E[\boldsymbol{\xi}])'], \quad \Sigma_{ij} = \text{Cov}(\xi_i, \xi_j)

$Matrix of covariances for multivariate random vector$

Multivariate Normal CF

f_{\boldsymbol{\xi}}(\boldsymbol{t}) = \exp\{i\boldsymbol{t}'\boldsymbol{\mu} - \frac{1}{2}\boldsymbol{t}'\boldsymbol{\Sigma}\boldsymbol{t}\}

$Characteristic function of multivariate normal distribution$

Linear Transformation of Vector

\text{If } \boldsymbol{\eta} = \boldsymbol{A}\boldsymbol{\xi} + \boldsymbol{b}, \text{ then } E[\boldsymbol{\eta}] = \boldsymbol{A}E[\boldsymbol{\xi}] + \boldsymbol{b}, \quad \text{Cov}(\boldsymbol{\eta}) = \boldsymbol{A}\text{Cov}(\boldsymbol{\xi})\boldsymbol{A}'

$Mean and covariance under linear transformation$

Conditional Expectation

E[\eta|\xi = x] = \begin{cases} \sum_j y_j P(\eta = y_j | \xi = x) & \text{discrete} \\ \int y p_{\eta|\xi}(y|x) dy & \text{continuous} \end{cases}

$Expected value of η given ξ = x$

Law of Total Expectation

E[\eta] = E[E[\eta|\xi]]

$Expectation equals expected value of conditional expectation$

Discrete Expectation

Continuous Expectation

General Form (Stieltjes Integral)

Expectation of Function

Linearity of Expectation

Product of Independent Variables

Variance Definition

Computational Formula

Standard Deviation

Variance of Linear Transformation

Variance of Sum

Variance of Independent Sum

Covariance Definition

Computational Covariance

Correlation Coefficient

Covariance Properties

Bilinearity of Covariance

Independence Condition

Markov Inequality

Chebyshev Inequality

Cauchy-Schwarz Inequality

Jensen Inequality

Hölder Inequality

Raw Moments

Central Moments

Moment Relationships

Skewness Coefficient

Kurtosis Coefficient

Moment Generating Function

MGF Moment Formula

Characteristic Function Definition

Discrete Characteristic Function

Continuous Characteristic Function

Linear Transformation

Independence Property

Moment Generation from CF

Inversion Formula

Fourier Inversion

Bernoulli Distribution Ber(p)

Binomial Distribution B(n,p)

Poisson Distribution P(λ)

Geometric Distribution Geo(p)

Uniform Distribution U[a,b]

Exponential Distribution Exp(λ)

Normal Distribution N(μ,σ²)

Gamma Distribution Γ(α,β)

Joint Expectation

Covariance Matrix

Multivariate Normal CF

Linear Transformation of Vector

Conditional Expectation

Law of Total Expectation

📐 Key Properties & Theorems

Fundamental Properties

Key Relationships

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