Independent Random Variables

Study guide: Exam P-1

Random Variables $X$ and $Y$ are Independent if knowledge of one provides no information about the other. Formally, for all values $x$ and $y$ , the joint distribution factors into the product of the marginals.

$F(x, y) = F_X(x) \cdot F_Y(y)$

$f(x,y) = f_X(x)\cdot f_Y(y)$

Independence implies Covariance is zero ( $\text{Cov}(X,Y) = 0$ ), though the converse is not always true
For independent random variables: $E[XY] = E[X]\,E[Y]$ and $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$
The Central Limit Theorem applies to sums of independent and identically distributed (i.i.d.) random variables

Verifying Independence $X$ and $Y$ have joint density $f(x,y) = 2$ for $0 < x < y < 1$ (and 0 elsewhere). Are $X$ and $Y$ independent?

Answer The marginal densities are $f_X(x) = 2(1-x)$ and $f_Y(y) = 2y$ . Their product is $4y(1-x)$ , but $f(x,y) = 2 \neq 4y(1-x)$ in general, so $X$ and $Y$ are not independent.