Moments of the exponential law on the half-line

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypothesis used by the change-of-variables lemma cited below. For every real $\lambda >0$ and every integer $k\ge 0$ the function $r\mapsto r^k{e}^{-\lambda r}$ is Lebesgue integrable on $(0,\mathrm{\infty })$ and

${\int }_0^{\mathrm{\infty }}{r}^k{e}^{-\lambda r}dr=\frac{k!}{{\lambda }^{k+1}}\mathrm{.}$

The assertion is an identity for the Lebesgue integral of a nonnegative Borel function; no probabilistic interpretation and no normalised density are asserted. In particular no statement is made about any other exponent, about complex $\lambda$, or about negative powers.

• Only the half-line is used. The change of variables is applied to the open half-line $(0,\mathrm{\infty })$, which is an open subset of $\mathbb{R}$; no endpoint value of the integrand at $0$ or at $+\mathrm{\infty }$ is used, and the equality is an equality in $[0,\mathrm{\infty }]$.
• Scaling convention. The exponent $k+1$ in the denominator and the factorial $k!$ in the numerator are fixed by the convention that the unscaled integral is $\mathrm{\Gamma }(k+1)=k!$; in particular the mean of the normalised density $r\mapsto \lambda e^{-\lambda r}$ is $1\mathrm{/}\lambda$.