elemprob-page20 - The multivariate distribution function of...

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Unformatted text preview: The multivariate distribution function of (X, Y ) is defined by FX,Y (x, y ) = P(X ≤ x, Y ≤ y ). In the continuous case, this is x y −∞ −∞ fX,Y (x, y )dy dx, and so we have f (x, y ) = ∂2F (x, y ). ∂x∂y The extension to n random variables is exactly similar. We have d b P(a ≤ X ≤ b, c ≤ Y ≤ d) = fX,Y (x, y )dy dx, c a or P((X, Y ) ∈ D) = fX,Y dy dx D 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 ) = fX,Y (x, y )dy dx. {x<y } If one has the joint density of X and Y , one can recover the densities of X and of Y : ∞ fX (x) = ∞ fX,Y (x, y )dy, fY (y ) = −∞ fX,Y (x, y )dx. −∞ A multivariate random vector is (X1 , . . . , Xr ) with P(X1 = n1 , . . . , Xr = nr ) = n! pn 1 · · · pn r , r n1 ! · · · nr ! 1 where n1 + · · · + nr = n and p1 + · · · pr = 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 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, pX,Y (x, y ) = P(X = x, Y = y ) = P(X = x)P(Y = y ) = pX (x)pY (y ). In other words, the joint density pX,Y factors. In the continuous case, b d fX,Y (x, y )dy dx = P(a ≤ X ≤ b, c ≤ Y ≤ d) = P(a ≤ X ≤ b)P(c ≤ Y ≤ d) a c b = d fX (x)dx a fY (y )dy. c 20 ...
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.

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