Find nth central moment for exponential distribution

Question

I am trying to figure out what the nth central moment is for the exponential distribution. Here is the formula for the nth moment:

$$ \mathop{\mathbb{E}}{[x^n]} =\dfrac{n!}{\lambda^n} $$

My question: what is the continuous formula for the exponential distribution for the nth central moment?

Answer 1

Generally, when $X$ is any random variable with a defined positive integral moment $\mu_n(X) = E[X^n],$ then (a) $\mu_1 = \mu = E[X]$ is defined and (b) its central moment of order $n$ is given by the Binomial Theorem as a linear combination of moments

$$\mu^\prime_n(X) = E[(X - \mu)^n] = E\left[\sum_{i=0}^n \binom{n}{i}\mu^{n-i}X^i \right] = \sum_{i=0}^n \binom{n}{i}\mu^{n-i}\mu_i(X).$$

This formula can be inverted to express moments in terms of central moments, too. See Stuart & Ord, Kendall's Advanced Theory of Statistics, Volume 1.

---

Perhaps the easiest way to obtain the moments of an exponential random variable is through its moment generating function (mgf), here computed for a standard exponential distribution with density function $f_X = e^{-x}$ defined on the positive real numbers:

$$\begin{aligned} \phi_X(t) &=\sum_{i=0}^\infty \frac{(-1)^i\mu_i\,t^i}{i!} = \sum_{i=0}^\infty \frac{-(1)^iE[X^i]\,t^i}{i!} = E\left[\sum_{i=0}^\infty \frac{(-tX)^i}{i!}\right] = E\left[e^{-tX}\right]\\& = \int_0^\infty e^{-tx} e^{-x}\,\mathrm d x = \int_0^\infty e^{-(t+1)x}\,\mathrm dx = \frac{1}{-(t+1)}e^{-(t+1)x}\bigg|_0^\infty = (1+t)^{-1}\\ &= \sum_{j=0}^\infty (-1)^i t^i. \end{aligned}$$

The first line is the entire point of the mgf, the second line is the elementary calculation of the integral, and the last line applies the Binomial Theorem.

Comparing these absolutely convergent power series term by term gives

$$\mu_i = i!.$$

When $\lambda$ is a rate parameter, $1/\lambda$ is a scale parameter. The units calculus tells us to multiply $\mu_i$ by $(1/\lambda)^i$ in this case, producing the formula $\mu_i = i!/\lambda^i$ quoted in the question. You may plug this in for $\mu_i$ in the general formula for central moments, bearing in mind $\mu = \mu_1 = 1/\lambda.$