{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Week5-Class4-Multivariate Distr.

Week5-Class4-Multivariate Distr. - Multivariate...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Multivariate Distributions Enrique Campos-N´ a˜nez The George Washington University ApSc 116
Image of page 1

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

View Full Document Right Arrow Icon
Multivariate Data: Height, Weight, and Fat Motivating multivariate distributions
Image of page 2
Table of Contents Bivariate Distributions Conditional Distributions Multivariate Distributions
Image of page 3

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

View Full Document Right Arrow Icon
Bivariate Distributions I The joint probability distribution of two r.v.’s is called a bivariate distribution . I The distribution is discrete if both r.v.’s are discrete. I Definition: The joint probability function (or joint p.f.) of X and Y is the function f such that ( x , y ) ∈ < 2 , f ( x , y ) = Pr ( X = x , Y = y ) . I If ( x , y ) is not one of the possible pairs of values of ( X , Y ) then f ( x , y ) = 0. Always, f ( x , y ) 0. I If the sequence { ( x i , y i ) : i = 1 , 2 , . . . } contains all possible values of ( X , Y ), then X i =1 f ( x i , y i ) = 1 . I Also, Pr ( X , Y ) A ⊂ < 2 = X ( x i , y i ) A f ( x i , y i ) .
Image of page 4
Exercise Let X 1 , X 2 represent the number of points in two dice, and let X = min { X 1 , X 2 } , Y = max { X 1 , X 2 } . What is the joint probability function of X and Y ?
Image of page 5

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

View Full Document Right Arrow Icon
Continuous Bivariate Distributions I We say that X and Y have a continuous joint distribution if there is a function f ( x , y ) 0 that is defined for all ( x , y ) ∈ < 2 and such that A ⊂ < 2 , Pr [( X , Y ) A ] = Z A Z f ( x , y ) dxdy . I The function f ( x , y ) is the joint probability density function or joint p.d.f., of X and Y . I In analogy to the univariate case, f ( x , y ) must satisfy f ( x , y ) 0 , -∞ < x , y < , and Z -∞ Z -∞ f ( x , y ) dxdy = 1 . I Note that if X and Y have a continuous joint distribution, then 1. Any point, or infinite sequence of points, in < 2 has a probability 0, and 2. Any one-dimensional curve in < 2 has probability 0.
Image of page 6
Mixed Bivariate Distributions This covers all other cases. Example Select one person at random from members of the class and let X = 1 if the person is male, X = 2 if the person is female, and let Y be the weight in kilograms of the person.
Image of page 7

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

View Full Document Right Arrow Icon
Bivariate Distribution Functions I Definition: The joint distribution function , or joint d.f., of r.v.’s X and Y is the function F such that ( x , y ) ∈ < 2 , F ( x , y ) Pr ( X x , Y y ) . Result. For all a b , and c d , we have Pr ( a < X b , c < Y d ) = F ( b , d ) - F ( a , d ) - F ( b , c ) + F ( a , c ) . a b c d
Image of page 8
Bivariate Distribution Functions II Definition. The function F 1 ( x ) Pr ( X x ) = lim y →∞ Pr ( X x , Y y ) = lim y →∞ F ( x , y ) , and is called the marginal d.f. of X .
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern