Joint Distribution of $Y_r$ and $Y_s$

The joint p.d.f. of $Y_r$ and $Y_s$, ($r<s$) is found by integrating over the other $(n-2)$ variables. Then

$$f_{Y_r,Y_s}(y_r,y_s) = \int^\infty_{y_s}\dots \int^\infty_{y_{n-1}} \left\{ \int^{y_s}_{... ...y} \int^{y_2}_{-\infty} n! \prod^n_{i=1}f(y_i)dy_1\dots dy_{r-1}\right] \right.$$

$$\quad\quad .\,dy_{r+1}\dots dy_{s-1}\left. \right\} dy_n\dots dy_{s+1}$$ (5.5)

The order of integration is first over $y_1,\,y_2,\,\dots ,$ to $y_{r-1}$, then over $y_{r+1}, \ y_{r+2}, \dots ,$ to $y_{s-1}$ and finally over $y_n, \ y_{n-1}, \dots ,$ to $y_{s+1}$. The limits of integration are obtained from the inequalities

$$\begin{eqnarray*} -\infty < y_1 < y_2 < & \dots & < y_{r-1} < y_r,\\ y_r < y_... ...< y_s,\\ \infty > y_n > y_{n-1} > & \dots & > y_{s+1} > y_s. \end{eqnarray*}$$

In order to integrate (5.8) we use methods similar to (5.3) and (5.4) together with

$${\int^{y_s}_{y_r} \dots \int^{y_{r+3}}_{y_r} \int^{y_{r+2}}_{y_r} \prod^{s-1}_{i=r+1} f(y_i)dy_{r+1}dy_{r+2} \dots dy_{s-1} }$$

$$= \int^{y_s}_{y_r} \dots\int^{y_{r+3}}_{y_r} [F(y_{r+2})-F_(y_r)] f_X(y_{r+2})\prod^{s-1}_{i=r+3} f(y_i )dy_{r+2}\dots dy_{s-1}$$

$$= \int^{y_s}_{y_r} \dots \int^{y_{r+4}}_{y_r} \frac{[F(y_{r+3})- F(y_r)]^2}{2!} f(y_{r+3}) \prod^{s-1}_{i=r+4} f(y_i)dy_{r+3} \dots dy_{s- 1}$$

$$= \frac{[F(y_s)-F(y_r)]^{s-r-1}}{(s-r-1)!}, \ \ \mbox{on simplification}.$$ (5.6)

Thus we get

$$\begin{eqnarray*} f_{Y_r,Y_s}(y_r,y_s) & = & n!f(y_r)f(y_s) \int^\infty_{y_s} ... ...y_r)]^{s-r-1}}{(s-r-1)!} \ \frac{[1-F(y_s)]^{n-s}}{(n-s)!} \ . \end{eqnarray*}$$

Hence the pdf of $(Y_r, Y_s)$ is given by

We now give an alternative derivation of the special case of (5.10) where $r=1$, $s=n$. In this method we first find the joint cumulative distribution function, then derive the joint pdf from it by differentiation.

The joint cdf of $Y_1$ and $Y_n$ is $P(Y_1 \leq y_1, Y_n \leq y_n)$. We use the result that

$$\{Y_n \leq y_n\}=\{Y_1 \leq y_1 \cap Y_n \leq y_n\} \cup \{Y_1 > y_1 \cap Y_n \leq y_n\}$$

where the two events on the RHS are mutually exclusive, to get an expression for our term of interest which is the first term on the RHS. So

$$\begin{eqnarray*} P(Y_n \leq y_n)&=&P(Y_1 \leq y_1 \cap Y_n \leq y_n)+P(Y_1 \geq... ...(\mbox{all }n \mbox{ obs. are between } y_1 \mbox{ and }y_n) \ .\end{eqnarray*}$$

So

$$F_{Y_1,Y_n}(y_1,y_n)=\left[F(y_n)\right]^n \, - \, \left[F(y_n)-F(y_1) \right]^n,y_1 \leq y_n$$

So the joint pdf is

$$\frac{\partial}{\partial y_1}\frac{\partial}{\partial y_n}\left[F_{Y_1,Y_n} (y_1,y_n)\right] = \frac{\partial}{\partial y_1}\left[n\left(F(y_n)\right)^{n-1}f(y_n) \, - \, n\left[F(y_n)- F(y_1)\right]^{n-1}f(y_n)\right]$$

$$= \, 0 \, + \, n(n-1)\left[F(y_n)-F(y_1)\right]^{n-2}f(y_n)f(y_1)$$ (5.7)

which is (5.10) with $r=1$ and $s=n$.