Conditional Probability The conditional probability that the outcome ‘y’ is ‘bj’ given that the…

Conditional Probability The conditional probability that the outcome ‘y’ is ‘bj’ given that the outcome x is ak is defined as (,) ( ) ( ) XY k j Y X j k X k P ab P ba P a ? ? = , assuming that, ( ) 0 1.3.4 P ak > or in short, ( ) (, ) ( ) p x y P yx p x = Two events x = ak and y = bj are said to be statistically independent if, ( , ) ( ). ( ) P ab P a Pb XY k j X k Y j = So in this case, ( ). ( ) ( ) ( ( ) Xk Yj Y X j k Y j X k P a Pb P ba P b P a ? ?= = ) 1.3.5 Two ensembles X and Y will be statistically independent if and only if the above condition holds for each element in the joint sample space, i.e., for all j-s and k-s. So, two ensembles X and Y are statistically independent if all the pairs (ak, bj) are statistically independent in the joint sample space. V