Question

It is said that a random variable X has the Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\,{\bf{and}}\,{\bf{\alpha }}\) if X has a continuous distribution for which the pdf\({\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right)\) is as follows

\(\begin{array}{l}{\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right){\bf{ = }}\frac{{{\bf{\alpha }}{{\bf{x}}_{\bf{0}}}^{\bf{\alpha }}}}{{{{\bf{x}}^{{\bf{\alpha + 1}}}}}}\,{\bf{,x}} \ge {{\bf{x}}_{\bf{0}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{ = }}\,{\bf{0}}\,\,{\bf{,x < }}{{\bf{x}}_{\bf{0}}}\end{array}\)

Show that if X has this Pareto distribution, then the random variable\({\bf{log}}\left( {{\bf{X|}}\,{{\bf{x}}_{\bf{0}}}} \right)\)has the exponential distribution with parameter α.

Short answer

The random variable\({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter α.

Step-by-step solution

Given information

X has the Pareto distribution with parameters \({x_0}\,{\rm{and}}\,\alpha \). We need to prove that the random variable \({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter α.

Proof of random variable log(X|xo) has the exponential distribution with parameter α.

X follows has the Pareto distribution with parameters \({x_0}\,{\rm{and}}\,\alpha \) then the CDF of X is given by

\(P\left( {X \le x} \right) = 1 - {\left( {\frac{{{x_0}}}{x}} \right)^\alpha }\)

Let \(Y = {\rm{log}}\left( {\frac{X}{{{x_0}}}} \right)\)then in order to show that the random variable \(\log \left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter α we consider the following

\(\begin{array}{l}P\left( {Y \le y} \right) = P\left( {{\rm{log}}\left( {\frac{X}{{{x_0}}}} \right) \le y} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = P\left( {\frac{X}{{{x_0}}} \le {e^y}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = P\left( {X \le {x_0}{e^y}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - {\left( {\frac{{{x_0}}}{{{x_0}{e^y}}}} \right)^\alpha }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1 - {e^{ - \alpha y}}\end{array}\)

This indicates that the random variable \({\rm{log}}\left( {X|\,{x_0}} \right)\) has the exponential distribution with parameter α.