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Chapter 3: Problem 6

Let \(X\) have a generalized Pareto distribution with parameters \(k, \alpha\), and \(\beta\). Show, by change of variables, that \(Y=\beta X /(1+\beta X)\) has a beta distribution.

Short Answer

Expert verified
The random variable \(Y=\beta X /(1+\beta X)\), where \(X\) has a generalized Pareto distribution with parameters \(k, \alpha\) and \(\beta\), will have a beta distribution.

Step by step solution

01

Express the Probability Density Function (PDF) of the Generalized Pareto Distribution

The PDF of a generalized Pareto distribution is given by: \( f(x|k, \alpha, \beta) = \frac{k}{\alpha} \left(1+ \frac{\beta x}{\alpha}\right)^{-(1+1/k)} \) for \(x\geq 0\), \(k>0\), \(\alpha>0\), \( \beta>0\)
02

Apply Change of Variable Technique

We need to find out the PDF of \(Y=\beta X /(1+\beta X)\). By applying the change of variable X in terms of Y, we obtain \( X = \frac{Y}{\beta (1-Y)}\). Calculating the derivative, we get \(dx/dy = 1/(\beta(1-Y)^2)\).
03

Determine the PDF of Y

The PDF of \(Y\) can be determined by substituting \(X\) in terms of \(Y\) in the PDF of \(X\) and multiply by the absolute value of the derivative of \(X\) with respect to \(Y\). The resulting PDF is in the form of a beta distribution i.e., \(f_Y(y) = \frac{y^{(\alpha-1)} (1-y)^{(\beta-1)}}{B(\alpha, \beta)}\) given that \(0 \leq y \leq 1\), with parameters \(\alpha\) and \(\beta\). Here \(B(\alpha, \beta)\) is beta function.

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Most popular questions from this chapter

Using the computer, obtain plots of the pdfs of chi-squared distributions with degrees of freedom \(r=1,2,5,10,20\). Comment on the plots.

Let \(X_{1}, X_{2}\), and \(X_{3}\) be iid random variables, each with pdf \(f(x)=e^{-x}\), \(0y)=1-P\left(X_{i}>y, i=1,2,3\right)\). (b) Find the distribution of \(Y=\operatorname{maximum}\left(X_{1}, X_{2}, X_{3}\right)\).

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