# Confidence Interval for Pareto Distribution A random variable is

Confidence Interval for Pareto Distribution
A random variable is said to have probability density function
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Step 1
$\stackrel{^}{k}>x$ if and only if
and the probability of that is $\left(\mathrm{Pr}\left({X}_{1}>x\right){\right)}^{n}$
$Pr\left({X}_{1}>x\right)={\int }_{x}^{\mathrm{\infty }}\frac{\alpha {k}^{\alpha }}{{u}^{\alpha +1}},du={\left(\frac{k}{x}\right)}^{\alpha },$
so
$Pr\left(\stackrel{^}{k}>x\right)={\left(\frac{k}{x}\right)}^{n\alpha }.$
Thus,
$Pr\left({x}_{1}<\stackrel{^}{k}<{x}_{2}\right)={\left(\frac{k}{{x}_{1}}\right)}^{n\alpha }-{\left(\frac{k}{{x}_{2}}\right)}^{n\alpha }.$
$Pr\left(Ak<\stackrel{^}{k}
$Pr\left(A<\frac{\stackrel{^}{k}}{k}
$Pr\left(\frac{1}{B}<\frac{k}{\stackrel{^}{k}}<\frac{1}{A}\right)=\cdots$
$Pr\left(\frac{\stackrel{^}{k}}{B}
This gives you a confidence interval for k if $\alpha$ is known. Since $\alpha$ is not known, there is more work to do. (You have to choose A and B to get you the probablities that you want.)