An alternative to the lognormal distribution for modeling highly skewed populations is

Chapter 4, Problem 22

(choose chapter or problem)

An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters \(\theta \text { and } r\). The probability density function is

\(f(x)= \begin{cases}\frac{r \theta^{r}}{x^{r}+1} & x \geq \theta \\ 0, & x<\theta\end{cases}\)

The parameters \(\theta \text { and } r\) may be any positive numbers. Let  be a random variable with this distribution.

a. Find the cumulative distribution function of .

b. Assume \(r>1\). Find \(\mu_{X}\)

c. Assume \(r>2\). Find \(\sigma_{X}^{2}\)

d. Show that if \(r \leq 1, \mu_{X}\) does not exist.

e. Show that if \(r \leq 2, \sigma_{X}^{2}\) does not exist.

Equation Transcription:

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Text Transcription:

\theta and  r

f(x)= {\frac{r \theta^{r}}{x^{r}+1} & x \geq \theta \\0, & x<\theta

\theta and  r

r>1

\mu_{X}

r>2

\sigma_{X}^{2}

r \leq 1, \mu_{X}

r \leq 2, \sigma_{X}^{2}