Joint Distribution Functions/ part4

The probability mass functions pX(x) and pY(y) are also known as the marginal distributions of X and Y, respectively.

In general, the probability mass function of a random variable Xi can be derived from the joint probability distribution of the set of discrete random variables X1 , X2, .... , Xn .

P(X1 , X2, .... , Xn )=P(X1=x1 , X2=x2, .... , Xn=xn)

And this function can be written as follows:

P(X1 , X2, .... , Xn )=P(X1=x1)P( X2=x2) ....P( Xn=xn)

Two random variables are independent if and only if their joint probability distribution function is simply the product of the simple probability distribution for each random variable. That is, the random variables X and Y are independent, if and only if

pX,Y(x, y) = pX(x) pY(y)


The above joint probability distribution of X and Y is tabulated as follows:

(x, y) pX,Y (x, y)
A A
(0, 0) 2/9
(0, 1) 1/3
(0, 2) 1/15
(1, 0) 2/9
(1, 1) 2/15
(2, 0) 1/45

Note that 2/9 + 1/3 + 1/15 + 2/9 + 2/15 + 1/45 = 1.