memLosycecyjz

2022-09-25

Let ${Y}_{1},{Y}_{2},...,{Y}_{n}$ be independent, uniformly distributed random variables on the interval $\left[0,\theta \right]$. density function of ${Y}_{\left(n\right)}$

Cremolinoer

Density function of maximum:
${g}_{\left(n\right)}\left(y\right)=n\left[F\left(y\right){\right]}^{n-1}f\left(y\right)$
Density function of minimum:
${g}_{\left(1\right)}\left(y\right)=n\left[1-F\left(y\right){\right]}^{n-1}f\left(y\right)$
${Y}_{i}$ are uniformly distributed on $\left[0,\theta \right]:$
$F\left(y\right)=\frac{y-0}{\theta -0}=\frac{y}{\theta }$ if $0\le y\le \theta$
$f\left(y\right)=\frac{1}{\theta -0}=\frac{1}{\theta }$ if $0\le y\le \theta$
We need to determine the probability density function of the maximum:
${g}_{\left(n\right)}\left(y\right)=n\left[F\left(y\right){\right]}^{n-1}f\left(y\right)=n\left(\frac{y}{\theta }{\right)}^{n-1}\frac{1}{\theta }=\frac{n{y}^{n-1}}{{\theta }^{n}}$
Result:
${g}_{\left(n\right)}\left(y\right)=\frac{n{y}^{n-1}}{{\theta }^{n}}$

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