Topic 12: Independent Random Variables

We have previously defined statistical independence of two events A and b in F. We will now use that definition to define independence of random variables X and Y.

Definition $\qquad$ Two random variables X and Y on (S,F,P) are statistically independent if the events {X ∈ A}, and {Y ∈ B} are independent ∀A,B ∈ F. i.e.

$P(\{X\in A\}\cap\{Y\in B\})=P(X\in A)P(Y\in B) \quad\forall A,B\in\mathcal F$

There is an alternative definition of independence for random variables that is often used. We will show that X and Y are independent iff

$f_{XY}(x,y)=f_X(x)f_Y(y)\quad\forall x,y\in\mathbb R$

First assume that X and Y are independent and let A = (-∞,x], B = (-∞,y]. Then,

\begin{align} F_{XY}(x,y) &= P(X\leq x,Y\leq y) \\ &= P(X\in A,Y\in B) \\ &= P(X\in A)P(Y\in B) \\ &= P(X\leq x)P(Y\leq y) \\ &= F_X(x)F_Y(y) \\ \Rightarrow f_{XY}(x,y) &= f_X(x)f_Y(y) \end{align}

Now assume that f$_{XY}$(x,y) = f$_X$(x)f$_Y$(y) ∀x,y ∈ R. Then, for any A,B ∈ B(R)

\begin{align} P(X\in A,Y\in B) &= \int_A\int_Bf_{XY}(x,y)dydx \\ &=\int_A\int_Bf_X(x)f_Y(y)dydx \\ &=\int_Af_X(x)dx\int_Bf_Y(y)dy \\ &= P(X\in A)P(Y\in B) \end{align}

Thus, X and Y are independent iff f$_{XY}$(x,y) = f$_X$(x)f$_Y$(y).