Note16 RV

• 笔记

• Function of RV

  • LOTUS: \(E[f(X)] = \sum_{x}f(x)P [X = x].\)

• Variance and Covariance

  • Variance定义：For a r.v. X with expectation \(E[X] = µ\), the variance of X is defined to be \(Var(X) = E[(X − µ)^2]\)
    • The square root \(σ(X) :=\sqrt{Var(X)}\) is called the standard deviation of X
  • 定理：For a r.v. X with expectation \(E[X] = \mu\), we have \(Var(X)=E[X^2]-\mu^2\)

• Variance Computation

• Sum of Independent Random Variables

  • 定理 ：For independent random variables X,Y , we have E[XY] = E[X]E[Y].
  • 定理：For independent random variables X,Y , we have Var(X +Y) = Var(X) +Var(Y).

• Covariance and Correlation

  • 表达式E[XY]−E[X]E[Y]是X和Y之间关联的度量，被称为协方差

  • 定义Covariance：\(Cov(X,Y) = E[(X - \mu_X )(Y - \mu_Y )] = E[XY]-E[X]E[Y]\)

    • 如果X，Y相互独立，那么Cov(X,Y) = 0。然而，逆命题不成立。
    • Cov(X,X) = Var(X)
    • Covariance is bilinear; i.e., for any collection of random variables \(\{X_1,...,X_n\},\{Y_1,...,Y_m\}\) and fixed constants \(\{a_1,\dots,a_n\},\{b_1,\dots,b_m\}\)
      • \(Cov(\sum_{i=1}^n a_{i}X_{i},\sum_{j=1}^{m} b_jY_j) = \sum_{i=1}^n \sum_{j=1}^{m} a_ib_jCov(X_i,Y_j).\)
    • For general random variables X and Y, Var(X +Y) = Var(X) +Var(Y) +2Cov(X,Y).

  • 定义Correlation：Suppose X and Y are random variables with σ(X) > 0 and σ(Y) > 0. Then, the correlation of X and Y is defined as \(Corr(X,Y) =\frac{Cov(X,Y)}{σ(X)σ(Y)}\)

  • 定理：For any pair of random variables X and Y with σ(X) > 0 and σ(Y) > 0, −1 ≤ Corr(X,Y) ≤ +1.
    • Corr比Cov更有用，因为它的值总是介于 -1 与 1之间
    • proof 表明，如果 \(Corr(X,Y) = \pm1\),则 \(Y=aX+b\), a,b 是常数，这表明二者线性相关，一者确认另一者同时确定

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