Derive the moment generating function of each of the random variables below, and use it to define the mean and variance of the random variable.

(a) \(f(x)=.2 e^{-.2 x} I_{(0, \infty)}(x)\)

(b) \(f(x)=2 x I_{(0,1)}(x)\)

(c) \(f(x)=.3^{x} \cdot 7^{1-x} I_{\{0,1\}}(x)\)