This is an important technical skill, so let’s drill it. For each of these, plot the transformation and then derive the range and the density of the new distribution:

1. Let \(X\sim\text{Unif}(0,\,1)\) and consider \(Y=1/X\);
2. Let \(X\) have density \(f_X(x)=42x^5(1-x)\) for \(0<x<1\) and consider \(Y=X^3\);
3. Let \(X\sim\text{Exponential}(\lambda)\) and consider \(Y=4X+3\);
4. Let \(X\) have cdf \(F_X(x)=(x+1)^3/8\) for \(-1<x<1\) and consider \(Y=1-X^2\);
5. Let \(X\sim\text{N}(0,\,1)\) and consider \(Y=|X|\).