Summary 6

# Summary 6 - Chapter 6 summary Jointly Distributed Random...

This preview shows pages 1–2. Sign up to view the full content.

Chapter 6 summary Jointly Distributed Random Variables Basic problem: Let X and Y be random variables. Consider the pair ( X,Y ). This is not a random variable because it isn’t a real-valued function (since its values are in R 2 , not R ). However, if D is any subset of R 2 , we’d like to know P (( X,Y ) D ), i.e. the probability that ( X,Y ) takes values in D . This is achieved in case X and Y are discrete by using the joint probability mass function p X,Y ( x,y ) and in case X and Y are continuous by using the joint probability density function f X,Y ( x,y ). These are each described next. Joint probability mass function: If X and Y are discrete random variables, the joint probability mass function is deﬁned to be p X,Y ( x,y ) = P ( X = x and Y = y ) . For any subset D of the set of possible values of ( X,Y ), P (( X,Y ) D ) = X ( x,y ) D p X,Y ( x,y ) . The joint mass function has the property that it is 0 and x,y p X,Y ( x,y ) = 1 if the sum is taken over all possible pairs of values that ( X,Y ) can take. Joint probability density function: If X and Y are continuous random variables, the joint prob- ability density function is denoted by f X,Y ( x,y ). It has the property that it is 0 and for subsets D of R 2 for which we can deﬁne P (( X,Y ) D ) we have P (( X,Y ) D ) = Z D Z f X,Y ( x,y ) dA. The support of the joint probability density function is basically the set of ( x,y ) where f X,Y ( x,y ) > 0. More precisely, a point is in the support of f X,Y if for every disk centered at that point, the probability that ( X,Y ) is in that disk is strictly positive. In writing the above double integral as an iterated double integral, it is important to draw the region bounded by the intersection of

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/05/2011 for the course MATH 351 taught by Professor Moumen,f during the Spring '08 term at George Mason.

### Page1 / 3

Summary 6 - Chapter 6 summary Jointly Distributed Random...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online