# 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:

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

## 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 $a > n$.

 $\ds \expect {X^n}$ $=$ $\ds a b^a \int_b^\infty x^{n - a - 1} \rd x$ $\ds$ $=$ $\ds a b^a \bigintlimits {\dfrac {x^{n - a} } {n - a} } b \infty$ Primitive of Power $\ds$ $=$ $\ds \dfrac {a b^a} {n - a} \paren {\lim_{x \mathop \to \infty} x^{n - a} - b^{n - a} }$ $\ds$ $=$ $\ds a b^a \paren {0 - \dfrac {b^{n - a} } {n - a} }$ for $n - a < 0$, $x^{n - a} \mathop \to 0$ as $x \mathop \to \infty$ $\ds$ $=$ $\ds \dfrac {a b^n} {a - n}$

$\Box$

Now take $a = n$.

 $\ds \expect {X^n}$ $=$ $\ds a b^a \int_b^\infty x^{a - a - 1} \rd x$ $\ds$ $=$ $\ds a b^a \bigintlimits {\ln x } b \infty$ Primitive of Reciprocal $\ds$ $=$ $\ds a b^a \paren {\lim_{x \mathop \to \infty} \ln x - \ln b }$ $\ds$ $\to$ $\ds \infty$ Logarithm Tends to Infinity

$\Box$

Finally, take $a < n$.

 $\ds \expect {X^n}$ $=$ $\ds a b^a \int_b^\infty x^{n - a - 1} \rd x$ $\ds$ $=$ $\ds a b^a \bigintlimits {\dfrac {x^{n - a} } {n - a} } b \infty$ Primitive of Power $\ds$ $=$ $\ds \dfrac {a b^a} {n - a} \paren {\lim_{x \mathop \to \infty} x^{n - a} - b^{n - a} }$ $\ds$ $\to$ $\ds \infty$ for $n - a > 0$, $x^{n - a}$ increases without bound as $x \mathop \to \infty$

$\blacksquare$