Statistics Assignment Help With Joint Probability Law

3.5 Joint Probability Law:

Two random variables X and Y are said to be jointly distributed if they are defined on the same probability space. The sample points consist of 2 tuples. If the joint probability function is denoted by PXY(x,y) then the probability of a certain event E is given by

PXY(x,y) = P[(X,Y) Є E]

(X,Y) is said to belong to E, if in the 2 dimensional space the 2 tuples lie in the borel set B, representing the event E.

3.5.1 Joint probability mass function and marginal and conditional probability function:

Let X and Y be random variables on a sample space S with respective image sets X(S) = { x1, x2,……. xn} and Y(S) ={ y1, y2,……. Ym}. We make the product set

X(S) *Y(S) = { x1, x2,……. xn}*={ y1, y2,……. Ym}

Into a probability space by defining the probability of the ordered pair (xi, yj). the function p on X(S) *Y(S) defined by

Is called the joint probability function of X and Y and is usually represented in the form of the following table:

Y X	y1	y2	y3	...….	yj	………	ym	Total
x1	p11	p12	p13	……..	p1j	………	p1m	p1.
x2	p21	p22	p23	……..	p2j	……..	p2m	p2.
x3	p31	p32	p33	……..	p3j	……..	p3m	p3.
. .	. .	. .	. .		. .		. .	. .
xi	pi1	pi2	pi3	……..	pij	……..	pim	pj.
. .	. .	. .	. .		. .		. .	. .
xn	pn1	pn2	pn3	……..	pnj	……..	pnm	pn.
Total	p.1	p.2	p.3	……..	p.j	……..	p.m	1

Two random variables X and Y are said to be independent if

two random variables

Otherwise they are said to be independent.

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