Question

Let x be a random variable with pareto distribution with parameter α . determine the probability distribution function of the random variable y = ln ⁡ ( x ) . what kind of distribution does y have?

Answer 1

I'm assuming the support of
`X` is
`x\ge1`, judging by the usage of
`Y=\ln X`. The CDF of
`X` is


`F_X(x)=\mathbb P(X\le x)=\begin{cases}1-\frac1{x^\alpha}&\text{for }x\ge1\\\\0&\text{otherwise}\end{cases}`

and so


`F_Y(y)=\mathbb P(Y\le y)=\mathbb P(\ln X\le y)=\mathbb P(X\le e^y)=F_X(e^y)`

`\implies F_Y(y)=\begin{cases}1-e^(-\alpha y)&\text{for }y\ge0\\\\0&\text{otherwise}\end{cases}`

`\implies f_Y(y)=(\mathrm d)/(\mathrm dy)F_Y(y)=\begin{cases}\alpha e^(-\alpha y)&\text{for }y>0\\0&\text{otherwise}\end{cases}`

which is the PDF of an exponential distribution with rate parameter
`\alpha`.