The problem is as follows:
Suppose we have have independent random variable $Y_1, Y_2, … ,Y_n$, and they are uniformly distributed over the closed interval [0,1]. If $V$ and $W$ are the smallest and largest, respectively of the $Y_i$'s, what is the joint PDF of $V$ and $W$?
I am approaching this problem by first trying to find the pdf's of the marginal and the conditional, and then multiplying them to find the joint. So I am trying to find $f_W(w)$ and $f_{v|w}(v|w)$.
Finding $f_W(w)$ is pretty straightforward.
$W = \max(Y_1, Y_2, Y_3, …, Y_n)$
$W \leq w$ iff $Y_k \leq w, \forall k$
$P(Y_k \leq w) = w$, because $Y_k$ is uniformly distributed on [0,1]
$\therefore P(W \leq w) = P(Y_1 \leq w)P(Y_2 \leq w)…P(Y_n \leq w) = w^n$
$\therefore F_w(w) = w^n, 0 \leq w \leq 1$
$\therefore f_w(w) = nw^{n-1}, 0 \leq w \leq 1$
I am unsure of how to continue from here to find $f_{v|w}(v|w)$
Best Answer
For the Max- and Min distribution we can derive the following two statements:
For the joint pdf every $\texttt{independent}$ random variable $Y_i$ must be smaller than $w$.
For the joint pdf every $\texttt{independent}$ random variable $Y_i$ must be larger than $v$.
That means $P(v<Y_i<w)=w-v \qquad \forall \ \ i =\{1,2,...,n \} $
Now use the independence to find the joint distribution when $n$ variables are involved.