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L14posted_rev1

# L14posted_rev1 - Multiple Continuous Random Variables Joint...

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Multiple Continuous Random Variables Joint probability density function f X,Y ( x, y ) describes pair ( X, Y ) . f X,Y ( x, y ) 0 Z x = -∞ Z y = -∞ f X,Y ( x, y ) dydx = 1 P (( X, Y ) A ) = Cumulative joint distribution function F X,Y ( x, y ) = 1

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Marginal Densities The p.d.f. of X alone is f X ( x ) = The p.d.f. of Y alone is f Y ( y ) = 2
Conditional Densities The conditional p.d.f. of X given Y = y is f X | Y ( x | y ) = f X,Y ( x, y ) f Y ( y ) The conditional p.d.f. of Y given X = x is f X | Y ( y | x ) = f X,Y ( x, y ) f X ( x ) 3

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Expectations Same rules as for discrete ran- dom variables apply but integrals replace sums. For instance, if u ( x, y ) is a function then the ex- pected value of u ( X, Y ) is de- fined as E [ u ( X, Y )] = 4
.

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Visualizing Joint Densities Left plot is joint p.d.f (surface in space), right shows level lines. 5
Visualizing Joint Densities Level lines qualitatively show the correlation between X, Y. 6

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Propagation of Errors Determining p.d.f of a function W = u ( X, Y ) requires a multiple integration. Cannot always be obtained analytically. But possi-
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L14posted_rev1 - Multiple Continuous Random Variables Joint...

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