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
Otherwise they are said to be independent.
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