Independent random variables

Summary

• $X$ and $Y$ are two random variables. We say that $X$ and $Y$ are independent random variables if the events $\{ X≤a \}$ and $\{ Y≤b \}$ are independent events for all real numbers $a,b$ .
• Thus, $X$ and $Y$ are independent random variables if and only if

$$F\left( x,y \right)=F_X\left( x \right)F_Y\left( y \right)$$

• for all $x,y$ .
• If $X$ and $Y$ are both discrete random variables or $X$ and $Y$ are both continuous random variables then this condition is equivalent to

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

Example

• $f\left( x,y \right)=4xy$ for $0≤x≤ 1$ and $0≤y≤ 1$ . Then $f_X\left( x \right)=2x$ and $f_Y\left( y \right)=2y$ . Since $f\left( x,y \right)=f_X\left( x \right)f_Y(y)$ for all $x,y$ , $X$ and $Y$ are independent random variables.
• If $X_1,…,X_n$ are $n$ random variables then we say that they are mutually / pairwise independent if $\{ X_1≤a_1 \}$ , $\{ X_2≤a_2 \},…,\{ X_n≤a_n \}$ are mutually / pairwise independent events for all real numbers $a_1,…,a_n$ .