On Lab 7, we learned how to derive the distribution of a transformation. This is an important technical skill that crops up often in mathematical probability, so let’s get our reps in:

1. Let \(X\sim\text{Unif}(0,\,1)\), and find the range and density of \(Y=e^X\);
2. Let \(X\sim\textrm{Unif}(0,\,1)\), and find the range and density of \(Y=\sqrt{X}\);
3. Let \(X\sim\text{Unif}(0,\,1)\), and find the range and density of \(Y=X^2\);
4. Let \(X\sim\text{Gamma}(a,\,b)\), and find the range and density of \(Y=1/X\);
5. Let \(Z\sim\text{N}(0,\,1)\), and find the range and density of \(Z^2\).