Suppose $X_1,\ldots,X_m\sim E(\mu,\sigma_1)$, $Y_1,\ldots,X_n\sim E(\mu,\sigma_2)$ are two independent random sample. If $W=\min(X_{(1)},Y_{(1)})$, how can find distribution $W$?
Note: $X_{(1)}$ is smallest order statistics in sample $X_1,\ldots,X_m$ and $Y_{(1)}$ is smallest order statistics in sample $Y_1,\ldots,Y_n$ and if ($X\sim E(\mu,\sigma)$ then f(x)=\frac{1}{\sigma}\displaystyle e^\frac{-(x-\mu)}{\sigma},\ x>\mu).