To show that the moment of order $n$ exists for the Pareto distribution if and only if $B > n$, we start by recalling the probability density function (pdf) of the Pareto distribution:

$f(x) = \frac{B a^B}{x^{B+1}}, \quad \text{for} \quad x \geq a, \quad \alpha > 0, \quad B > 0$ where $a$ is the scale parameter and $B$ is the shape parameter.

Step 1: Definition of the $n$-th moment

The $n$-th moment of a continuous random variable $X$ is defined as: $E[X^n] = \int_{a}^{\infty} x^n f(x) \, dx$

For the Pareto distribution, we substitute the pdf $f(x) = \frac{B a^B}{x^{B+1}}$ into this expression:

$E[X^n] = \int_{a}^{\infty} x^n \cdot \frac{B a^B}{x^{B+1}} \, dx$

Simplifying the integrand: $E[X^n] = B a^B \int_{a}^{\infty} x^{n - (B + 1)} \, dx = B a^B \int_{a}^{\infty} x^{n - B - 1} \, dx$

Step 2: Evaluate the integral

We now solve the integral: $\int_{a}^{\infty} x^{n - B - 1} \, dx$

This is a standard integral of the form $\int x^k \, dx$, where $k = n - B - 1$. The result of the integral is:

$\int_{a}^{\infty} x^k \, dx = \frac{x^{k+1}}{k+1} \Bigg|_a^{\infty}$

If $k + 1 < 0$ (or $n - B - 1 + 1 = n - B < 0$), then the upper limit $x = \infty$ leads to the term going to zero. This occurs when $B > n$. In this case:

$\frac{x^{k+1}}{k+1} \Bigg|_a^{\infty} = \frac{a^{n-B}}{B-n}$

Thus, the $n$-th moment is: $E[X^n] = B a^B \cdot \frac{a^{n-B}}{B-n} = \frac{B a^n}{B-n}$

Step 3: Condition for the moment to exist

For the $n$-th moment to exist, the integral must converge, which happens only if $B > n$. If $B \leq n$, the integral diverges, meaning the moment does not exist.

Conclusion:

The moment of order $n$ exists if and only if $B > n$.

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Would you like further clarification or any additional details on this topic? Here are some related questions to explore:

1. How does the Pareto distribution compare with other heavy-tailed distributions?
2. Can you derive the mean and variance of the Pareto distribution?
3. How is the Pareto distribution used in real-world scenarios?
4. What is the significance of the shape parameter $B$ in the Pareto distribution?
5. How does the Pareto distribution change for different values of $B$?

Tip: The Pareto distribution is widely used to model wealth distribution, where a small percentage of the population controls most of the wealth.