Conditional Probability Distribution

The conditional probability distribution of a discrete random variable X, given a second discrete random variable Y, is the probability distribution of X conditioned on the fact that Y assumes a value y. The conditional probability distribution of X given Y is denoted by p_X|Y(x|y) and is defined as

Conditional probabilities, like simple probabilities, also sum up to 1, i.e.,

If X and Y are independent random variables, then

i.e., the probability distribution of X is not affected by the value of Y, and vice-versa.

EX. The random variables X and Y have the following joint probability distribution:

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

The marginal distributions p_x(x) and p_y(y) can be calculated as follows:

= p_x,y(x,0) + p_x,y(x,1) + p_x,y(x,2)

p_x(0) = p_X,Y(0,0) + p_X,Y(0,1) + pX,Y(0,2)

= 2/9 + 1/3 + 1/15
= 28/45

px(1) = p_X,Y(1,0) + p_X,Y(1,1) + p_X,Y(1,2)

= 2/9 + 2/15
= 16/45

p_x(2) = p_X,Y(2,0) + p_X,Y(2,1) + p_X,Y(2,2)

= 1/45 

 

p_y(0) = p_X,Y(0,0) + p_X,Y(1,0) + p_X,Y(2,0)

= 2/9 + 2/9 + 1/45
= 7/15

p_y(1) = p_X,Y(0,1) + p_X,Y(1,1) + p_X,Y(2,1)

= 1/3 + 2/15
= 7/15

p_y(2) = p_X,Y(0,2) + p_X,Y(1,2) + p_X,Y(2,2)

= 1/15

The conditional probability of X given Y is calculated as follows:

p_x(0|0) = (15/7) p_X,Y(0,0) = (15/7) (2/9) = 10/21
p_x(1|0) = (15/7) p_X,Y(1,0) = (15/7) (2/9) = 10/21
p_x(2|0) = (15/7) p_X,Y(2,0) = (15/7) (1/45) = 1/21

p_x(0|1)= (15/7) p_X,Y(0,1) = (15/7) (1/3) = 5/7
p_x(1|1)= (15/7) p_X,Y(1,1) = (15/7) (2/15) = 2/7
p_x(2|1)= (15/7) p_X,Y(2,1) = (15/7) (0) = 0

p_x(0|2) = 15 p_X,Y(0,2) = 15 (1/15) = 1
p_x(1|2) = 15 p_X,Y(1,2) = 15 (0) = 0
p_x(2|2) = 15 p_X,Y(2,2) = 15 (0) = 0

The conditional probability of Y given X is calculated as follows:

p_y(0|0) = (45/28) p_X,Y(0,0) = (45/28)(2/9) = 5/14
p_y(1|0) = (45/28) p_X,Y(0,1) = (45/28)(1/3) = 15/28
p_y(2|0) = (45/28) p_X,Y(0,2) = (45/28)(1/15) = 3/28

p_y(0|1) = (45/16) p_X,Y(1,0) = (45/16)(2/9) = 5/8
p_y(1|1) = (45/16) p_X,Y(1,1) = (45/16)(2/15) = 3/8
p_y(2|1) = (45/16) p_X,Y(1,2) = (45/16)(0) = 0

p_y(0|2) = 45 p_X,Y(2,0) = 45 (1/45) = 1
p_y(1|2) = 45 p_X,Y(2,1) = 45 (0) = 0
p_y(2|2) = 45 p_X,Y(2,2) = 45 (0) = 0