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A member of the Pareto family of distributions (often used
Chapter 6, Problem 18E(choose chapter or problem)
A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution function given by
\(F(y)=\left\{0, y<\beta 1-\left(\frac{\beta}{y}\right)^{\alpha}, y \geq \beta\right.\)
where \(\alpha, \beta>0\).
a Find the density function.
b For fixed values of \(\beta \text { and } \alpha\), find a transformation \(G(U)\) so that \(G(U)\) has a distribution function of \(F\) when \(U\) has a uniform distribution on the interval .
c Given that a random sample of size 5 from a uniform distribution on the interval yielded the values
and
, use the transformation derived in part (b) to give values associated with a random variable with a Pareto distribution with \(\alpha=2, \beta=3\)
Equation Transcription:
Text Transcription:
F(y)=0, y<\beta 1-\left \beta y^\alpha, y \geq \beta
\alpha, \beta>0
\beta and \alpha
G(U)
G(U)
F
U
\alpha=2, \beta=3
Questions & Answers
QUESTION:
A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution function given by
\(F(y)=\left\{0, y<\beta 1-\left(\frac{\beta}{y}\right)^{\alpha}, y \geq \beta\right.\)
where \(\alpha, \beta>0\).
a Find the density function.
b For fixed values of \(\beta \text { and } \alpha\), find a transformation \(G(U)\) so that \(G(U)\) has a distribution function of \(F\) when \(U\) has a uniform distribution on the interval .
c Given that a random sample of size 5 from a uniform distribution on the interval yielded the values
and
, use the transformation derived in part (b) to give values associated with a random variable with a Pareto distribution with \(\alpha=2, \beta=3\)
Equation Transcription:
Text Transcription:
F(y)=0, y<\beta 1-\left \beta y^\alpha, y \geq \beta
\alpha, \beta>0
\beta and \alpha
G(U)
G(U)
F
U
\alpha=2, \beta=3
ANSWER:Solution:
Step 1 of 4:
The Pareto distribution function is given by
Where,