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Formula Reference

Digital Characteristics Formulas

Comprehensive collection of essential formulas for digital characteristics including expectation, variance, covariance, moments, characteristic functions, and their properties

8 Categories52 Formulas
Mathematical Expectation
Fundamental formulas for mathematical expectation and its properties
6 formulas

Discrete Expectation

E[ξ]=∑k=1∞xkp(xk)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[ξ]=∫−∞∞xp(x)dxE[\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[ξ]=∫−∞∞xdF(x)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(ξ)]={∑kg(xk)p(xk)discrete∫−∞∞g(x)p(x)dxcontinuousE[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[∑i=1nciξi+b]=∑i=1nciE[ξi]+bE\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

If ξ1,…,ξn independent, then E[ξ1⋯ξn]=E[ξ1]⋯E[ξn]\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

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

Variance as expectation of squared deviation from mean

Computational Formula

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

More convenient formula for calculating variance

Standard Deviation

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

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

Variance of Linear Transformation

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

Translation doesn't affect variance, scaling affects quadratically

Variance of Sum

Var(∑i=1nξi)=∑i=1nVar(ξi)+2∑1≤i<j≤nCov(ξi,ξj)\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

If ξ1,…,ξn independent, then Var(∑i=1nξi)=∑i=1nVar(ξi)\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

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

Measures how two variables jointly deviate from their respective means

Computational Covariance

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

More convenient formula for calculating covariance

Correlation Coefficient

ρ(ξ,η)=Cov(ξ,η)Var(ξ)⋅Var(η)\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

Cov(aξ+b,cη+d)=ac⋅Cov(ξ,η)\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

Cov(∑i=1mξi,∑j=1nηj)=∑i=1m∑j=1nCov(ξi,ηj)\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

ξ and η independent⇒Cov(ξ,η)=0\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(∣ξ∣≥ε)≤E[∣ξ∣]ε,ε>0P(|\xi| \geq \varepsilon) \leq \frac{E[|\xi|]}{\varepsilon}, \quad \varepsilon > 0

Upper bound for tail probability using first moment

Chebyshev Inequality

P(∣ξ−E[ξ]∣≥ε)≤Var(ξ)ε2,ε>0P(|\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[ξη]∣2≤E[ξ2]⋅E[η2]|E[\xi\eta]|^2 \leq E[\xi^2] \cdot E[\eta^2]

Fundamental inequality relating product expectation to second moments

Jensen Inequality

If g(x) is convex, then g(E[ξ])≤E[g(ξ)]\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[∣ξη∣]≤(E[∣ξ∣p])1/p(E[∣η∣q])1/q,1p+1q=1E[|\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

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

k-th moment about origin

Central Moments

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

k-th moment about the mean

Moment Relationships

c1=0,c2=Var(ξ),m1=E[ξ]c_1 = 0, \quad c_2 = \text{Var}(\xi), \quad m_1 = E[\xi]

Special cases of first and second moments

Skewness Coefficient

γ1=c3c23/2=E[(ξ−E[ξ])3](Var(ξ))3/2\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

γ2=c4c22−3=E[(ξ−E[ξ])4](Var(ξ))2−3\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

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

Generates all moments through derivatives at origin

MGF Moment Formula

If Mξ(t) exists, then E[ξk]=Mξ(k)(0)\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

fξ(t)=E[eitξ]=∫−∞∞eitxdF(x),t∈Rf_\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

fξ(t)=∑k=1∞eitxkp(xk)f_\xi(t) = \sum_{k=1}^{\infty} e^{itx_k} p(x_k)

Characteristic function for discrete random variables

Continuous Characteristic Function

fξ(t)=∫−∞∞eitxp(x)dxf_\xi(t) = \int_{-\infty}^{\infty} e^{itx} p(x) dx

Characteristic function for continuous random variables

Linear Transformation

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

Characteristic function under linear transformation

Independence Property

If ξ1,…,ξn independent and η=∑i=1nξi, then fη(t)=∏i=1nfξi(t)\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

If E[∣ξ∣n]<∞, then E[ξk]=fξ(k)(0)ik,0≤k≤n\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(x2)−F(x1)=lim⁡T→∞12π∫−TTe−itx1−e−itx2itfξ(t)dtF(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

If ∫−∞∞∣fξ(t)∣dt<∞, then p(x)=12π∫−∞∞e−itxfξ(t)dt\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[ξ]=p,Var(ξ)=p(1−p),f(t)=peit+(1−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[ξ]=np,Var(ξ)=np(1−p),f(t)=(peit+(1−p))nE[\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[ξ]=λ,Var(ξ)=λ,f(t)=eλ(eit−1)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[ξ]=1p,Var(ξ)=1−pp2,f(t)=peit1−(1−p)eitE[\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[ξ]=a+b2,Var(ξ)=(b−a)212,f(t)=eitb−eitait(b−a)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[ξ]=1λ,Var(ξ)=1λ2,f(t)=11−itλE[\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[ξ]=μ,Var(ξ)=σ2,f(t)=eitμ−σ2t22E[\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[ξ]=αβ,Var(ξ)=αβ2,f(t)=(1−itβ)−αE[\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(ξ,η)]={∑i∑jg(xi,yj)pijdiscrete∫∫g(x,y)p(x,y)dxdycontinuousE[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

Σ=E[(ξ−E[ξ])(ξ−E[ξ])′],Σij=Cov(ξi,ξj)\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

fξ(t)=exp⁡{it′μ−12t′Σt}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

If η=Aξ+b, then E[η]=AE[ξ]+b,Cov(η)=ACov(ξ)A′\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[η∣ξ=x]={∑jyjP(η=yj∣ξ=x)discrete∫ypη∣ξ(y∣x)dycontinuousE[\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[η]=E[E[η∣ξ]]E[\eta] = E[E[\eta|\xi]]

Expectation equals expected value of conditional expectation

📐 Key Properties & Theorems

Essential properties and relationships for digital characteristics

Fundamental Properties

  • • Linearity: E[aX + bY + c] = aE[X] + bE[Y] + c
  • • Independence: E[XY] = E[X]E[Y] if X ⊥ Y
  • • Variance scaling: Var(aX + b) = a²Var(X)
  • • Covariance bilinearity: Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z)

Key Relationships

  • • Variance formula: Var(X) = E[X²] - (E[X])²
  • • Correlation bound: |ρ(X,Y)| ≤ 1
  • • Independence ⇒ uncorrelated: X ⊥ Y ⇒ Cov(X,Y) = 0
  • • CF uniqueness: f(t) uniquely determines distribution

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