elemprob-fall2010-page35

elemprob-fall2010-page35 - for P ( { ω : X ( ω ) is in A...

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or P (( X,Y ) D ) = Z Z D f X,Y dy dx when D is the set { ( x,y ) : a x b,c y d } . One can show this holds when D is any set. For example, P ( X < Y ) = Z Z { x<y } f X,Y ( x,y ) dy dx. If one has the joint density of X and Y , one can recover the densities of X and of Y : f X ( x ) = Z -∞ f X,Y ( x,y ) dy, f Y ( y ) = Z -∞ f X,Y ( x,y ) dx. A multivariate random vector is ( X 1 ,...,X r ) with P ( X 1 = n 1 ,...,X r = n r ) = n ! n 1 ! ··· n r ! p n 1 1 ··· p n r r , where n 1 + ··· + n r = n and p 1 + ··· p r = 1. In the discrete case we say X and Y are independent if P ( X = x,Y = y ) = P ( X = x,Y = y ) for all x and y . In the continuous case, X and Y are independent if P ( X A,Y B ) = P ( X A ) P ( Y B ) for all pairs of subsets A,B of the reals. The left hand side is an abbreviation
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Unformatted text preview: for P ( { ω : X ( ω ) is in A and Y ( ω ) is in B } ) and similarly for the right hand side. In the discrete case, if we have independence, p X,Y ( x,y ) = P ( X = x,Y = y ) = P ( X = x ) P ( Y = y ) = p X ( x ) p Y ( y ) . In other words, the joint density p X,Y factors. In the continuous case, Z b a Z d c f X,Y ( x,y ) dy dx = P ( a ≤ X ≤ b,c ≤ Y ≤ d ) = P ( a ≤ X ≤ b ) P ( c ≤ Y ≤ d ) = Z b a f X ( x ) dx Z d c f Y ( y ) dy. 35...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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