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In this document topics covered which are Random Vectors or Multi-dimensional Random Variables, Joint Probability Mass Function, Definition, two continuously distributed rv’s , Conditional pdf .
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Random Vectors or Multi-dimensional Random Variables : A sequence of random variables X =(X 1 ,…,Xp), where each Xi is defined for the same sample space and probability measure, is called a random vector. Definition: Cumulative distribution function of a p dimensional random vector X =(X 1 ,X 2 ,…,Xp) is defined as F(x 1 ,…,xp)=P(X 1 x 1 ,…, Xpxp). We consider two-dimensional rv’s.
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With each point (xi,yj) (i,j=1,2,…) we associate a number p(xi,yj)=P(Xi=xi,Yj=yj) satisfying the following conditions (i) p(xi,yj)0, (ii)ii) p(xi,yj)= The function p(x,y) satisfying the above conditions is called the joint pmf of (X,Y). Marginal Probability Mass Function : The marginal pmf of the r v X is given by p (^) X (xi)=j p(xi , y (^) j); i=1,2,…
Definition: Joint cdf of X,Y is given by where x and y are any two real numbers. Obviously F(x,y) represents the probability that the rv X takes value less than or equal to x and Y takes value less than or equal to y. Marginal cdf of X is given by If X,Y are independently distributed then (i) p(y|x)=pY(y). (ii) F(x,y)=FX(x)FY(y). (verify)
i jx xy y i j i j F x y p x y , :
j X i i j F ( x ) F ( x , y )
Example : The joint pmf of (X,Y) is given by p(x,y): Find (i) k, (ii) marginal pmf of X, (iii) conditional pmf of Y given X=2, (iv) P(X=Y). X Y 0 1 2 0 1/24 k 1/ 1 2k 1/24 0 2 k 1/16 1/ 3 1/16 0 1/
The joint cdf of (X,Y) is given by The marginal pdf and cdf of X are given by Similarly we can obtain the marginal pdf and cdf of Y.
x y F x y f x y dy dx ( , )
F x F x f x y dy dx f x f x y dy x X X
Conditional pdf of Y given X=x is given by Two continuous rv’s (X,Y) are said to be independently distributed if f(x,y)=fX(x)fY(y) for all x,y. Then F(x,y)=FX(x)FY(y) (verify). Further, if X and Y are independently distributed then f(y|x)=fY(y) (verify).
f x f x y f y x X
