{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Joint_Distribution

# Joint_Distribution - Joint and Continuous In many...

This preview shows pages 1–7. Sign up to view the full content.

Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 1 Joint Distributions—Discrete and Continuous In many statistical investigations, one is frequently interested in studying the relationship between two or more random variables, such as the relationship between annual income and yearly savings per family or the relationship between occupation and hypertension.

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

View Full Document
Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 2 Discrete Variables For two discrete random variables X 1 and X 2 , the probability that X 1 will take the value x 1 and X 2 will take the value x 2 is written as P ( X 1 = x 1 , X 2 = x 2 ). Consequently, P ( X 1 = x 1 , X 2 = x 2 ) is the probability of the intersection of the events X 1 = x 1 and X 2 = x 2 . If X 1 and X 2 are discrete random variables, the function given by f ( x 1 , x 2 ) = P ( X 1 = x 1 , X 2 = x 2 ) for each pair of values ( x 1 , x 2 ) within the range of X 1 and X 2 is called the joint probability distribution or joint density function of X 1 and X 2 .
Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 3 The distribution of probability is specified by listing the probabilities associated with all possible pairs of values x 1 and x 2 , either by formula or in a table . A function of two variables can serve as the joint probability distribution of a pair of discrete random variables X 1 and X 2 if and only if its values, f ( x 1 , x 2 ), satisfy the conditions 1. f ( x 1 , x 2 ) 0 for each pair of values ( x 1 , x 2 ) within its domain; ∑ ∑ = 1 2 , 1 ) , ( . 2 2 1 x x x x f where the double summation extends over all possible pairs ( x 1 , x 2 ) within its domain.

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

View Full Document
Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 4 Joint distribution function: If X 1 and X 2 are discrete random variables, the function given by ∑∑ = = 1 2 ) , ( ) , ( ) , ( 2 2 1 1 2 1 x s x t t s f x X x X P x x F for - < x 1 < , - < x 2 < where f ( s , t ) is the value of the joint probability distribution of X 1 and X 2 at ( s , t ), is called the joint distribution function, or the joint cumulative distribution of X 1 and X 2 .
Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 5 Marginal distribution: If X 1 and X 2 are discrete random variables and f ( x 1 , x 2 ) is the value of their joint probability distribution at ( x 1 , x 2 ), the function given by = = = 2 ) , ( ) ( ) ( 2 1 1 1 1 1 x x x f x f x X P for each x 1 within the range of X 1 is called the marginal distribution of X 1 . Correspondingly, the function given by = = = 1 ) , ( ) ( ) ( 2 1 2 2 2 2 x x x f x f x X P for each x 2 within the range of X 2 is called the marginal distribution of X 2 .

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

View Full Document
Feb 23, 2011 Department of Mathematics, BITS Pilani, Goa Campus 6 Conditional probability distribution: Consistent with the definition of conditional probability of events when A is the event X 1 = x 1 and B is the event X 2 = x 2 , the conditional probability distribution of X 1 = x 1 given X 2 = x 2 is defined as 0 ) ( provided all for ) ( ) , ( ) | ( 2 2 1 2 2 2 1 2 1 1 = x f x x f x x f x x f Similarly, the conditional probability distribution of X 2 given X 1 = x 1 is defined as 0 ) ( provided all for ) ( ) , ( ) | ( 1 1 2 1 1 2 1 1 2 2 = x f x x f x x f
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern