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Unformatted text preview: Lecture Stat 302 Introduction to Probability  Slides 18 AD March 2010 AD () March 2010 1 / 13 Jointly Distributed Random Variables If both X and Y are continuous r.v., then their joint p.d.f. is a nonnegative function f ( x , y ) such that for any set C P f ( X , Y ) 2 C g = Z Z ( x , y ) 2 C f ( x , y ) dxdy . In particular, we have the following multivariate c.d.f. F ( a , b ) = P ( X & a , Y & b ) = Z a ¡ ∞ Z b ¡ ∞ f ( x , y ) dxdy . and, when we di/erentiate, we obtain f ( x , y ) = ∂ 2 ∂ x ∂ y F ( x , y ) . For X and Y are discrete r.v., then their joint p.m.f. is P ( X = x , Y = y ) = p ( x , y ) and P f ( X , Y ) 2 C g = ∑ ( x , y ) 2 C p ( x , y ) . AD () March 2010 2 ¡ 13 Independent Random Variables The r.v. X and Y are said to be independent if, for any sets A and B we have P ( X 2 A , Y 2 B ) = P ( X 2 A ) P ( Y 2 B ) It can be shown that X and Y are independent if and only if F ( x , y ) = F X ( x ) F Y ( y ) and f ( x , y ) = f X ( x ) f Y ( y ) that is the joint c.d.f. (resp. the joint p.d.f.) is the product of the marginal c.d.f.s (resp. the marginal p.d.f.s) AD () March 2010 3 / 13 Example The joint density of two r.v. X and Y is given by f ( x , y ) = & x e & ( x + y ) for x > 0 and y > otherwise....
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 Spring '10
 222
 Probability, Probability theory, probability density function, exp x exp

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