Raw Moment of Pareto Distribution

Theorem
Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.

Let $n$ be a strictly positive integer.

Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:


 * $\expect {X^n} = \begin {cases} \dfrac {a b^n} {a - n} & n < a \\ \text {does not exist} & n \ge a \end {cases}$

Proof
From the definition of the Pareto distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$

Where $\Img X \in \hointr b \infty$.

From the definition of the expected value of a continuous random variable:


 * $\ds \expect {X^n} = \int_b^\infty x^n \map {f_X} x \rd x$

First take $a > n$.

Now take $a = n$.

Finally, take $a < n$.