Computation of a mean

Question

The density of the RV X is $f(x)=x e^{-x} $ and I want to compute the following mean

$$\mathbb{E}[X^i e^{- \lambda X}].$$

Is $\mathbb{E}[X^i e^{- \lambda X}]=\int_0^{\infty} f(x) x^i e^{- \lambda x} \, dx $ the right way?

Answer 1

Yes. Note that your random variable $X$ follows a Gamma distribution, $f(x) = \begin{cases} x e^{-x} , \quad x> 0 \\ 0, \quad \text{otherwise} \end{cases} $.

Then $$\mathsf E \left [ X^i e^{-\lambda X} \right ] =\displaystyle \int_0^\infty f(x) \cdot x^i e^{-\lambda x} \, dx = \int_0^\infty x^{i + 1} e^{-x(\lambda + 1)} \, dx $$ which "resembles" another Gamma distribution.

$$\mathsf E \left [ X^i e^{-\lambda X} \right ] = \dfrac{ \Gamma(i + 2)}{(\lambda + 1)^{i+ 2}} \quad \blacksquare $$