Independent Random Variables and Conditional Distribution

Independent Random Variables

Two random variables are X and Y are independent if, for every A and B,
P(X ∈ A, Y ∈ B) = P(X ∈ A)P(Y ∈ B)

Two random variable X and Y which have joint pdf f_X,Y are independent if and only if f_X,Y(x, y) = f_X(x) f_Y(y) for all values of x and y.

Note that f_X(x) and f_Y(y) above are marginal distributions.

Let us take the following example of bivariate distribution.


f_X(0) = 1/3, f_X(1) = 2/3, f_Y(0) = 1/3 and f_Y(1) = 2/3. Here X and Y are independent because f_X(0) f_Y(0) = f(0, 0), f_X(1) f_Y(0) = f(1, 0), f_X(0) f_Y(1) = f(0, 1) and f_X(1) f_Y(1) = f(1, 1).

Conditional Distribution

Let X and Y be two discrete random variables. The conditional distribution of X given that we observe Y = y is expressed as
P(X = x| Y = y) = P(X = x, Y = y)/P(Y = y).

The conditional probability mass function is

if f_Y(y) > 0.

For continuous random variables, the conditional probability density function is

assuming that f_Y(y) > 0. Then,