Solution Found!
An alternative to the lognormal distribution for modeling
Chapter 4, Problem 22SE(choose chapter or problem)
An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters θ and r. The probability density function is
\(f(x)=\left\{\begin{array}{cc}
\frac{r \theta^{+}}{x^{*+1}} & x \geq \theta \\
x & x<0
\end{array}\right.
\)
The parameters and
may be any positive numbers. Let
be a random variable with this distribution.
a. Find the cumulative distribution function of .
b. Assume . Find \(\mu_{x}\).
c. Assume . 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:
Text transcription:
f(x)=\left\{\begin{array}{cc}
\frac{r \theta^{+}}{x^{*+1}} & x \geq \theta \\
x & x<0
\end{array}\right.
\mu_{x}
\sigma_{x}^{2}
r \leq 1, \mu_{x}
r \leq 2, \sigma_{x}^{2}
Questions & Answers
QUESTION:
An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters θ and r. The probability density function is
\(f(x)=\left\{\begin{array}{cc}
\frac{r \theta^{+}}{x^{*+1}} & x \geq \theta \\
x & x<0
\end{array}\right.
\)
The parameters and
may be any positive numbers. Let
be a random variable with this distribution.
a. Find the cumulative distribution function of .
b. Assume . Find \(\mu_{x}\).
c. Assume . 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:
Text transcription:
f(x)=\left\{\begin{array}{cc}
\frac{r \theta^{+}}{x^{*+1}} & x \geq \theta \\
x & x<0
\end{array}\right.
\mu_{x}
\sigma_{x}^{2}
r \leq 1, \mu_{x}
r \leq 2, \sigma_{x}^{2}
ANSWER:
Answer:
Step 1 of 5:
(a)
In this question we are asked to find the cumulative distribution function (CDF) of .
is a random variable with the following probability distribution function (PDF).
………….(1)
Where r and are parameters and this distribution is called Pareto distribution.
Therefore
We know the formula for calculating the cumulative distribution function (CDF) is:
………….(2)
Where is a probability distribution function of any random variable.
Since has defined in two separate intervals, the calculation of the CDF will involves two separate cases
If , then
will be
= 0 ………..(3)
If , then
will be
=
=
= using power rule of integration
=
= puting the limit
= 1 - (
= 1 - (
……….(4)
Hence cumulative distribution function (CDF) of
=
= 1 - (