# Thread: Order statistics question

1. ## Order statistics question

I'd like to know if my answers to the following question are correct:

Let $\displaystyle Y_1 \ Y_2, ..., \ Y_n$ be independent uniformly distributed random variables on the interval $\displaystyle [0, \ \theta]$.

a) Find the probability distribution function of $\displaystyle Y_{(n)} =\max(Y_1 \ Y_2, ..., \ Y_n)$

$\displaystyle =n(n-1)[F(y)]^{n-2}[f(y)]^2 + n [F(y)]^{n-1}f'(y)$

$\displaystyle =n(n-1) \left(\frac{1}{\theta}y \right) ^{n-2} \times \left(\frac{1}{\theta}\right)^2 + n(y)^{n-1}f'(y)$

$\displaystyle =n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2$

b) Find the density function of $\displaystyle Y_{(n)}$

$\displaystyle =n [F(y)]^{n-1}f(y)$

$\displaystyle n \left(\frac{1}{\theta}y \right)^{n-1} \times \left( \frac{1}{\theta} \right)$

c) the mean of $\displaystyle Y_{(n)}$

$\displaystyle \int_0^{\theta} y \times \left[ n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2\right] \ dy$

I'm not too sure about any of these answers.

2. Originally Posted by lllll
I'd like to know if my answers to the following question are correct:

Let $\displaystyle Y_1 \ Y_2, ..., \ Y_n$ be independent uniformly distributed random variables on the interval $\displaystyle [0, \ \theta]$.

a) Find the probability distribution function of $\displaystyle Y_{(n)} =\max(Y_1 \ Y_2, ..., \ Y_n)$

$\displaystyle =n(n-1)[F(y)]^{n-2}[f(y)]^2 + n [F(y)]^{n-1}f'(y)$

$\displaystyle =n(n-1) \left(\frac{1}{\theta}y \right) ^{n-2} \times \left(\frac{1}{\theta}\right)^2 + n(y)^{n-1}f'(y)$

$\displaystyle =n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2$

b) Find the density function of $\displaystyle Y_{(n)}$

$\displaystyle =n [F(y)]^{n-1}f(y)$

$\displaystyle n \left(\frac{1}{\theta}y \right)^{n-1} \times \left( \frac{1}{\theta} \right)$

c) the mean of $\displaystyle Y_{(n)}$

$\displaystyle \int_0^{\theta} y \times \left[ n(n-1)\left(\frac{1}{\theta}y \right)^{n-2} \times \left(\frac{1}{\theta}\right)^2\right] \ dy$

I'm not too sure about any of these answers.
a) How is a probablity distribution function different to a probability density function. Can you supply the definition you've been given - they would seem to be the same thing to me ......

b) Correct.

c) $\displaystyle E(Y_{(n)}) = \int_{0}^{\theta} y \, g(y) \, dy$ where g(y) is the pdf found in (b).