(a) Find the moment-generating function of a random variable \(X\) having the density function

\(f(x)=\frac{1}{8}\left(\begin{array}{c}3 \\ x\end{array}ight) I_{(0,1,2,3)}(x)\).

(Hint: Use of the binomial theorem may be helpful in finding a compact representation of this function.)

Use the MGF to calculate the first two moments of \(X\) about the origin. Calculate the variance of \(X\).

(b) Repeat

(a) using the density function

\(f(x)=\frac{1}{10} e^{-x / 10} I_{(0, \infty)}(x)\).