# Pareto distribution

So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found the maximum likelihood estimator of $k$,

$\widehat{k}=$min$(x_{1}, x_{2},..., x_{n})$ and it's distribution which was, $f_{\widehat{k}}(x) = \frac{ank^{\alpha n}}{x^{\alpha n + 1}}$

1) How would I go about finding the distribution of $\frac{k}{\widehat{k}}$

2) Find a 95% confidence interval using that? I know I need a function of $k$ and the estimator $\widehat{k}$ and that's why I am asking for it in 1).

I suggest you calculate the cumulative distribution function of $\frac{k}{\hat{k}}$.
• Yup then differentiate it to find the pdf right? But how would I set it up since $\frac{k}{\widehat{k}} = \frac{k}{min(x_{1}, ..., x_{n})}$ – Bobby May 26 '15 at 11:18
• For $x\geq k>0$, $P\left(\frac{k}{\hat{k}}\leq x\right) = P\left( \hat{k}\geq \frac{k}{x}\right) = P\left(\min (x_1,\dots,x_n)\geq \frac{k}{x}\right) = \dots$ – Augustin May 26 '15 at 11:22