Variance and Total Expectation¶
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Random Variables: Variance and Covariance¶
Variance¶
For a r.v. \(X\) with expectation \(E[X]=\mu\), the variance of \(X\) is defined to be: $$ Var(X)=E[(X-\mu)^2] $$ The square root \(\sigma(X)=\sqrt{Var(X)}\) is called the standard deviation of \(X\).
- The point of taking the square root of variance is to put the standard deviation “on the same scale” as the r.v. itself.
Theorem: For a r.v. \(X\) with expectation \(E[X]=\mu\), we have \(Var(X)=E[X^2]-{\mu}^2\)
Property: For any random variable \(X\) and constant \(c\), we have \(Var(cX)=c^2Var(X)\)
Sum of Independent Random Variable¶
Theorm: For independent variables \(X,Y\), we have \(E[XY]=E[X]E[Y]\).
Theorem: For independent random variables \(X,Y\), we have $$ Var(X+Y)=Var(X)+Var(Y) $$
It is very important to remember that neither of the above two results is true in general when X,Y are not independent
Covariance and Correlation¶
Covariance: The covariance of random variables \(X\) and \(Y\), denoted \(Cov(X,Y)\) is defned as $$ Cov(X,Y)=E[(X-\mu_x)(Y-\mu_Y)]=E[XY]-E[X]E[Y] $$
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If \(X,Y\) are independent, then \(Cov(X,Y)=0\). However, the converse is not true
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\(Cov(X,X)=Var(X)\)
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Covariance is bilinear: for any collection of random variables\(\left\{X_1,\cdots,X_n\right\},\left\{Y_1,\cdots,Y_n\right\}\) and fixed constants\(\left\{a_1,\cdots,a_n\right\},\left\{b_1,\cdots,b_n\right\}\), $$ Cov(\sum_{i=1}^na_iX_i,\sum_{j=1}^mb_jY_j)=\sum_{i=1}^n\sum_{j=1}^ma_ib_jCov(X_i,Y_j) $$
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For general random variables \(X\) and \(Y\):
While the sign of \(Cov(X,Y)\) is informative of how \(X\) and \(Y\) are associated, its magnitude is difficult to interpret. A statistic that is easier to interpret is correlation:
Correlation: Suppose \(X\) and \(Y\) are random variables with \(\sigma(X)>0\) and \(\sigma(Y)>0\). Then, the correlation of \(X\) and \(Y\) is defined as $$ Corr(X,Y)=\frac{Cov(X,Y)}{\sigma(X)\sigma(Y)} $$ Correlation is more useful than covariance because the former always ranges between −1 and +1, as the following theorem shows:
For any pair of random variable \(X\) and \(Y\) with \(\sigma(X)>0\) and \(sigma(Y)>0\), $$ -1\le Corr(X,Y)\le1 $$
创建日期: 2023年10月23日 09:55:35