If $X$ and $Y$ are independent and have the distribution function $F_X$ and $F_Y$, respectively. What is the distribution function of $\min(X,Y)$ and $\max(X,Y)$

The moment generating function (MGF) of standard normal distribution is given by

\(M_Z(t)=e^{\frac{t^2}{2}}\)

The MGF for \(N\left(\mu, \sigma^2\right)\) is given by

\(M_X(t)=e^{\mu t+ \frac{\sigma^2 t^2}{2}}\)

The moment generating function of Poisson distribution is given by

\(M(t)=e^{\lambda(e^t-1)}\)

Let \(Y=X_1+X_2+...+X_n\), where \(X_i\)s are independent and \(X_i\sim \mathrm{Poisson}(\lambda_i)\). We have \(Y\sim \mathrm{Possion}\left(\sum_{i=1}^{n}\lambda_{i}\right)\)