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321w2p1

# 321w2p1 - 4 Standard Deviation sd(X = X Properties of...

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4. Standard Deviation – sd(X) = X Properties of Standard Deviation: sd(c)=0 sd(aX+b) = |a| sd(x) Measures of Distribution Shape: 5. Skewness - measure of lack of symmetry = E [ X − X 3 ] X 3 when skewness=0 the distribution is symmetric

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6. Kurtosis -thickness of the distribution tails = E [ X − X 4 ] X 4 Measures of Association – For two or more r.v.s: 7. Covariance –measures the amount of linear dependence between two r.v.s Cov X,Y = E [ X − X ]= XY = E [ XY ]− X Y if Cov(X,Y)>0 then X and Y move in the same direction.
Properties of Covariance: When X and Y are independent, Cov(X,Y) = 0 Cov(aX+b, cY+d) = ac Cov(X,Y) | XY | ≤ X Y 8. Correlation Coefficient direction of relationship between two r.v.s Corr X,Y = Cov X,Y sx X sd Y = XY X Y = XY

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Properties of Correlation Coefficient: -1≤Corr(X,Y)≤1 Corr(X,Y)=0 implies X & Y are uncorrelated Corr(X,Y)=1 implies perfect positive linear relationship Corr(aX+b, cY+b) = Corr(X,Y) if ac>0 properties. 9. Conditional Expectation =conditional mean E [ Y|X = x ]= i = 1 n y i f Y|X y i |x
Properties of Conditional Expectation: E[g(X)|X] = g(X) for any function g(.) E[g(X)Y+h(X)|X] = g(X)E[Y|X] + h(X) E[Y|X] = E[Y] if X and Y are independent Law of Iterated Expectations - if we take the expectation of the mean of Y conditional on X, across all x i , then we will simply have the unconditional mean of Y E [ E [ Y|X ]]= i = 1 n E [ Y|X = x i ] P X = x i = E [ Y ]

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10. Conditional Variance – variance of the conditional distribution of one r.v. Given another Var Y|X = x = E [ Y 2 |X = x ]− E [ Y|X = x ] 2 = i = 1 n y i E [ Y|X = x ] 2 f Y|X y i |x if X & Y are independent, then Var(Y|X) = Var(Y)
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