Solution
= Solution
{c}
For an <exponential distribution> with rate $\lambda>0$, direct integration gives the <moment-generating function>
$$
M_X(t)=\int_0^\infty e^{tx}\lambda e^{-\lambda x}dx
=\boxed{\frac\lambda{\lambda-t},\quad t<\lambda.}
$$
It is finite in a neighborhood of zero, so its derivatives there give moments. Differentiating three times gives $M_X^{(3)}(t)=6\lambda/(\lambda-t)^4$, hence
$$
\boxed{\mathbb EX^3=M_X^{(3)}(0)=6/\lambda^3.}
$$