Raw Moment of Pareto Distribution
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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 $a > n$.
\(\ds \expect {X^n}\) | \(=\) | \(\ds a b^a \int_b^\infty x^{n - a - 1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a b^a \bigintlimits {\dfrac {x^{n - a} } {n - a} } b \infty\) | 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 $x \mathop \to \infty$ |
$\blacksquare$