Let $X_1, \dots , X_n$ be independently distributed with exponential density $$ f(x) = (2θ)^{−1}e^{−x/2θ}, x \geq 0 $$ and let the ordered $X$’s be denoted by $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$. It is assumed that $X_{(1)}$ becomes available first, then $X_{(2)}$, and so on, and that observation is continued until $X_{(r)}$ has been observed.
I try to find the joint distribution of $X_{(1)}, \dots , X_{(r)}$ and $\text{Cov}(X_{(r)}, X_{(s)})$ $(1 \leq r < s \leq n)$.
I've been thinking all the way down to get the joint function, but I've been struggling ever since.
$$ f(X(1), . . . , X(r)) = \int_{0}^{\infty} \int_{0}^{\infty} ... \int_{0}^{\infty} (2θ)^{−1}e^{−x/2θ} dx_1 dx_2 ... dx_n $$
I should like to have the benefit of your advice.
The joint density function for order statistics is complicated. Probability texts such as Sheldon Ross' books give the formula and a short derivation. Essentially the idea is to form the cdf first. $P(X_{(k)} \leq c)$ = probability that $k$ variables are less and $n-k$ variables are more than $c$. I suggest using that breakdown.
Your joint pdf should have $(2\theta)^{-n}e^{-x _1 /2\theta}e^{-x_2 /2\theta}...$