I am working on a mathematical stats assignment and I got stuck here.
Letting $X_1, X_2, ... ,X_n$ a random sample from uniform(0,$\theta$), and Y is n-th order statistic, I need to show that the pdf of Y is:
$p(y|\theta) = \frac{ny^{n-1}}{\theta^{n}}, 0<y<\theta$
I know that n-th order is $g_n(y_n) = nf(y_n)[\int_{-\infty}^{y_n}f(x)dx]^{n-1}.$
I plugged in the uniform distribution $f(x)=\frac{1}{\theta}$ and tried to solve the integral, and it didn't show the result I wanted. How should I approach this? Should I assign something else for $f(x)$?
Also, If I want to find the joint pdf f(y,θ) of Y and θ, which the prior pdf $\lambda(\theta)$ is given, I just multiply $p(y|\theta)$ to get the f(y,$\theta$), right?
Thanks in advance.
$$ F_Y(y|\theta) = \left( \mathbb{P}(X_1 \le y \right)^n=\left(\frac{y}{\theta} \right)^n, $$ hence $$ f_Y(y|\theta) = n \frac{ y ^{n-1}}{\theta ^n}, \quad y\in(0,\theta), $$ and using the multiplication rule, that is $\mathbb{P}(A \cap B) = \mathbb{P}(A|B)\mathbb{P}(B) $, $$ f_{Y, \theta}(y, \theta) = f_Y(y|\theta)\lambda(\theta) = n \frac{ y ^{n-1}}{\theta ^n}\lambda(\theta) $$