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.
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
Otherwise they are said to be independent.
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