NotesOct29

# NotesOct29 - Example Recall the example where X and Y are...

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Example : Recall the example where X and Y are discrete random variables with X denot- ing the number of defective welds, and Y the number of improperly tightened bolts produced per car. The joint and marginal pmfs are given by x i /y j 0 1 2 3 p X ( x i ) 0 .840 .030 .020 .010 .900 1 .060 .010 .008 .002 .080 2 .010 .005 .004 .001 .020 p Y ( y j ) .910 .045 .032 .013 1 1. Compute the conditional pmf of Y given X = 2; 2. compute the conditional mean E ( Y | X = 2) and the conditional variance Var( Y | X = 2).

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Conditional distributions in the continuous case Suppose that ( X, Y ) is continuous with a joint pdf f X,Y ( x, y ). Conditionally on Y = y , the random vari- able X is continuous. The conditional density is denoted by f X | Y ( x | y ). The conditional pdf is f X | Y ( x | y ) = f X,Y ( x, y ) f Y ( y ) . It is only legitimate to condition on the values y that are in the range of Y .
The conditional pmf of X given Y = y is f X | Y ( x | y ) = f X,Y ( x, y ) f Y ( y ) , the ratio of the joint pdf of X and Y and the marginal pdf of Y . The conditional pdf of Y given X = x is f Y | X ( y | x ) = f X,Y ( x, y ) f X ( x ) , the ratio of the joint pdf of

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Unformatted text preview: ( x ) , the ratio of the joint pdf of X and Y and the marginal pdf of X . Conditional expectation and conditional variance Conditional expectation E ( X | Y = y ) is the ex-pectation with respect to the conditional dis-tribution: E ( X | Y = y ) = Z ∞-∞ xf X | Y ( x | y ) dx. Conditional variance Var( X | Y = y ) is the vari-ance with respect to the conditional distribu-tion: Var( X | Y = y ) = E ( X 2 | Y = y )-( E ( X | Y = y )) 2 where E ( X 2 | Y = y ) = Z ∞-∞ x 2 f X | Y ( x | y ) dx. Example : Let ( X,Y ) be a continuous random vector with a joint pdf f X,Y ( x,y ) = 15 xy 2 , ≤ x,y ≤ 1 ,y ≤ x. • Compute the conditional densities f X | Y and f Y | X ; • Compute the conditional mean and the con-ditional variance of Y given X = x for < x < 1....
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• '10
• SAMORODNITSKY
• Probability theory, conditional pmf, conditional variance var

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