elemprob-fall2010-page34

# elemprob-fall2010-page34 - Z ∞ ce-x 1 2 e-2 x dx = c 6 so...

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14 Multivariate distributions We want to discuss collections of random variables ( X 1 ,X 2 ,...,X n ), which are known as random vectors. In the discrete case, we can deﬁne the density p ( x,y ) = P ( X = x,Y = y ). Remember that here the comma means “and.”” In the continuous case a density is a function such that P ( a X b,c Y d ) = Z b a Z d c f ( x,y ) dy dx. An example. If f X,Y ( x,y ) = ce - x e - 2 y for 0 < x < and x < y < , what is c ? Answer. We use the fact that a density must integrate to 1. So Z 0 Z x ce - x e - 2 y dy dx = 1 . Recalling multivariable calculus, this is
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Unformatted text preview: Z ∞ ce-x 1 2 e-2 x dx = c 6 , so c = 6. The multivariate distribution function of ( X,Y ) is deﬁned by F X,Y ( x,y ) = P ( X ≤ x,Y ≤ y ). In the continuous case, this is Z x-∞ Z y-∞ f X,Y ( x,y ) dy dx, and so we have f ( x,y ) = ∂ 2 F ∂x∂y ( x,y ) . The extension to n random variables is exactly similar. We have P ( a ≤ X ≤ b,c ≤ Y ≤ d ) = Z b a Z d c f X,Y ( x,y ) dy dx, 34...
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