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 \(a_1,…,a_n\) .