The moment-generating function of the random variable \(Y\) is given by \(M_{Y}(t)=(1-.25 t)^{-3}\) for \(t<4\).

(a) Find the mean and variance of the random variable \(Y\).

(b) Is the PDF of \(Y\) skewed? Why or why not?

(c) It is known that the moment generating function of the PDF \(f(x)=\frac{1}{\beta^{2} \Gamma(\alpha)} x^{\alpha-1} e^{-x / \beta} I_{(0, \infty)}(x)\) is given by \(M_{x}(t)=(1-\beta t)^{-\alpha}\) for \(t<\beta^{-1}\). The \(\Gamma(\alpha)\) in the preceding expression for the pdf is known as the gamma function, which for integer values of \(\alpha\) is such that \(\Gamma(\alpha)=(\alpha-1) !\). Define the exact functional form of the probability density function for \(Y\), if you can.