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]

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(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

joint probability law

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

joint probability mass function

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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