NotesNov8

# NotesNov8 - ρ = 2 a Find the mean and the variance of the...

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Properties of the bivariate normal distribu- tion : Let ( X, Y ) be jointly normal with the means μ X and μ Y , variances σ 2 X and σ 2 Y , and correlation ρ . Marginal distributions of X and Y : X N ( μ X , σ 2 X ) and Y N ( μ Y , σ 2 Y ) .

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Conditional distributions : given X = x , the conditional distribution of Y is again normal with the (conditional) mean E ( Y | X = x ) = ρ σ Y σ X ( x - μ X ) + μ Y and (conditional) variance Var( Y | X = x ) = 1 - ρ 2 σ 2 Y .
Given Y = y , the conditional distribution of X is again normal with the (conditional) mean E ( X | Y = y ) = ρ σ X σ Y ( y - μ Y ) + μ X and (conditional) variance Var( X | Y = y ) = 1 - ρ 2 σ 2 X .

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Linear combinations : for any numbers a, b the linear combination Z = aX + bY is, again normal with the mean EZ = X + Y and the variance Var Z = a 2 σ 2 X + b 2 σ 2 Y + 2 abρσ X σ Y .
Example : A portfolio consists of 100 units of stock A and 200 units of stock B. Let X and Y be monthly returns on single units of stocks A and B respectively. Suppose that ( X, Y ) are jointly normal with the means μ X = \$5, μ Y = \$3, standard deviations σ X = \$2, σ Y = \$1 and correlation

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Unformatted text preview: ρ = . 2. ( a ) Find the mean and the variance of the monthly portfolio return. ( b ) Find the probability of a monthly loss of more than \$100 on this portfolio. ( c ) Given that in a particular month a return on a unit of stock B was \$2, ﬁnd the probability that the entire portfolio returned a loss. Further properties of the bivariate Gaussian distribution Let X and Y be jointly normal distribution with the means μ X and μ Y , variances σ 2 X and σ 2 Y , and correlation ρ . • Suppose that the correlation ρ = 0. Then the joint pdf f X,Y ( x,y ) = f X ( x ) f Y ( y ) and so X and Y are independent. Conclusion : For jointly normal random vari-ables zero correlations imply independence. This is NOT true without the assumption of joint normality ....
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