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var(X)=E[(X−E[X])2]=E[X2]−(E[X])2var(X|Y=y)=E[(X−E[X|Y=y])2|Y=y] |
Law of total variance
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var(X)=E[var(X|Y)]+var(E[X|Y])var(X1+⋯+XN)=E[var(X1+⋯+XN|N)]+var(E[X1+⋯+XN|N])=E[N]⋅var(X)+(E[X])2⋅var(N) |
Covaraince
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cov(X,Y)=E[(X−E[X])⋅(Y−E[Y])]=E[XY]−E[X]⋅E[Y]cov(aX+b,Y)=a⋅cov(X,Y)cov(X,Y+Z)=cov(X,Y)+cov(X,Z) |
Correlation coefficient
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ρ(X,Y)=cov(X,Y)σXσY−1≤ρ≤1 |
Sum of random variables
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var(X1+X2)=var(X1)+var(X2)+2cov(X1,X2)var(X1−X2)=var(X1)+var(X2)−2cov(X1,X2) |
Source: MITx 6.041x, Lecture 12, 13.
# posted by rot13(Unafba Pune) @ 8:55 PM
