This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 10/21/2010 for the course STAT Stat302 taught by Professor 222 during the Spring '10 term at The University of British Columbia.
 Spring '10
 222
 Probability

Click to edit the document details