Independence of Continuous Random Variables

Recall that two discrete random variables $X$ and $Y$ are independent if and only if their joint PMF factors as $p(x,y)=p_X(x)\,p_Y(y)$ for all $x,y.$ The same idea applies in the continuous case, with PDFs replacing PMFs.

Two continuous random variables $X$ and $Y$ are independent if their joint PDF satisfies

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

for all $x,y\in\mathbb{R}.$

Intuitively, knowing the value of one variable provides no information about the other — the joint behavior is completely determined by the two marginal distributions. If this equation fails for some $(x,y),$ then $X$ and $Y$ are dependent.

For instance, suppose $X$ and $Y$ have joint PDF

$$f(x,y)=\begin{cases} x+y & 0\le x\le 1,\ 0\le y\le 1 \\ 0 & \text{otherwise}\end{cases}$$

with marginals $f_X(x)=x+\tfrac12$ and $f_Y(y)=y+\tfrac12$ on $[0,1].$ Then

$$f_X(x)\,f_Y(y)=\left(x+\tfrac12\right)\left(y+\tfrac12\right)=xy+\tfrac{x}{2}+\tfrac{y}{2}+\tfrac14,$$

which is not equal to $x+y.$ Therefore $X$ and $Y$ are dependent.