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Variance

Definition (Variance)

For a discrete RV XX with range X\mathscr{X}, pmf pp and expected value μ\mu, its Variance is Var(X):=E[(x−μ)2]=∑x∈X(x−μ)2 p(x)Var(X):=E[(x-\mu)^2]=\sum_{x\in\mathscr{X}}(x-\mu)^2 \ p(x)

Proposition

For a RV X∈L1X\in \mathscr{L}^{1} with mean μ\mu, Var(X)=E[x2]−μ2Var(X)=E[x^2]-\mu^2

Theorem

Var(X)=0  ⟺  X is a constant RVVar(X)=0\iff X \text{ is a constant RV}

Theorem

Var(aX+b)=a2Var(X)Var(aX+b)=a^2Var(X)

Theorem (Independence of Variance)

If X1,…,XnX_{1},\dots,X_{n} are Independent random variables then Var(X1+⋯+Xn)=∑i=1nVar(Xi)\text{Var}(X_{1}+\dots+X_{n})=\sum_{i=1}^{n}\text{Var}(X_{i})or equivalently if E[Xi]=0,∀i\mathbb{E}[X_{i}]=0,\forall i:E[(X1+⋯+Xn)2]=∑i=1nE[Xi2]\mathbb{E}[(X_{1}+\dots+X_{n})^{2}]=\sum_{i=1}^{n}\mathbb{E}[X_{i}^{2}]

Linked from

Monte Carlo Method

Gaussian Rate Distortion Function

Gaussian Source Maximizes Rate Distortion Function

Gain of Transform Coding

Karhunen-Loeve Transform

Optimal Bit Allocation

Discrete-Time Memoryless Gaussian Channel (AWGN Channel)

Gaussian Noise Minimizes Capacity of Additive-Noise Channel

Upper Bound on Channel Capacity

Closed-loop Predictor Coefficients

Wide Sense Stationary Process

Law of Large Numbers

Conditional Variance

Variance

Summary of MATH 895

Gaussian Random Variable

Log-Sum-Exp Trick

Principal Component Analysis