Consider \(X\sim\textrm{Gamma}(\alpha,\,\beta)\).

1. Find \(M_X(t)=E[e^{tX}]\), the moment generating function of \(X\). For what values of \(t\) is it defined?
2. Use the moment-generating function to compute \(E(X)\).
3. Use the moment-generating function to compute \(\text{var}(X)\).
4. If \(c>0\), what is the distribution of \(Y=cX\)?