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 fX,Y are independent if and only if fX,Y(x, y) = fX(x) fY(y) for all values of x and y.
Note that fX(x) and fY(y) above are marginal distributions.
Let us take the following example of bivariate distribution.
fX(0) = 1/3, fX(1) = 2/3, fY(0) = 1/3 and fY(1) = 2/3. Here X and Y are independent because fX(0) fY(0) = f(0, 0), fX(1) fY(0) = f(1, 0), fX(0) fY(1) = f(0, 1) and fX(1) fY(1) = f(1, 1).
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 fY(y) > 0.
For continuous random variables, the conditional probability density function is
assuming that fY(y) > 0. Then,