91Ó°ÊÓ

Referring to exercise 16 of sec.5.7, there is a Pareto distribution with the parameters \({x_0}\) and \(\alpha \)where, \({x_0} > 0\;and\;\alpha > 0\).

The p.d.f of the Pareto distribution is\({f_w}\left( w \right) = \frac{{\alpha {\theta ^\alpha }}}{{{w^{\alpha + 1}}}}\;;w \in \left( {\theta ,\alpha } \right)\).

Now let’s consider X is a random variable with the probability density function of\({f_X}\left( x \right) = \lambda {e^{ - \lambda x}};x > 0\).

And another random variable, Y, is defined as \(Y = c{e^x}\).

The cumulative distribution of Y can be written as,

\(\begin{aligned}{}F\left( y \right) &= \Pr \left( {Y \le y} \right)\\ &= \Pr \left( {c{e^x} \le y} \right)\\ &= \Pr \left( {{e^x} \le \frac{y}{c}} \right)\\ &= \Pr \left( {x \le \ln \left( {\frac{y}{c}} \right)} \right)\end{aligned}\)

Therefore, the c.d.f of Y is \(F\left( y \right) = {F_X}\left( {\ln \left( {\frac{y}{c}} \right)} \right)\)

The probability density function of Y is calculated as,

\(\begin{aligned}{}f\left( y \right) &= \frac{d}{{dy}}\left( {{F_Y}\left( y \right)} \right)\\ &= \frac{d}{{dy}}\left( {{F_X}\left\{ {\ln \left( {\frac{y}{c}} \right)} \right\}} \right)\\& = {f_x}\left( {\ln \left( {\frac{y}{c}} \right)} \right) \times \frac{1}{{\frac{y}{c}}} \times \frac{1}{c}\\ &= \frac{c}{y} \times \frac{1}{c} \times \lambda \times {e^{ - \lambda \ln \left( {\frac{y}{c}} \right)}}\\ = \frac{\lambda }{y}{e^{\ln {{\left( {\frac{y}{c}} \right)}^{ - \lambda }}}}\\ &= \frac{\lambda }{y} \times {\left( {\frac{y}{c}} \right)^{ - \lambda }}\\ &= \frac{\lambda }{y}{\left( {\frac{c}{y}} \right)^\lambda }\end{aligned}\)

So, the p.d.f of Y is \(f\left( y \right) = \frac{{\lambda {c^\lambda }}}{{{y^{\lambda + 1}}}}\).

Since,\(x > 0\).

So, there can be concluded that,

\(\begin{aligned}{}\left( {0 < x < \infty } \right) = {e^0} < {e^x} < \infty \\ = 1 < {e^x} < \infty \\ = c < c{e^x} < \infty \\ = c < y < \infty \end{aligned}\)

Therefore, the p.d.f of Y is \(f\left( y \right) = \frac{{\lambda {c^\lambda }}}{{{y^{\lambda + 1}}}};\;c < y < \infty \).

By comparing the probability distribution function of Y with the Pareto distribution,

\(y \sim Pareto\left( {c,\lambda } \right)\).