2. Let f and F be the pdf and cdf of a continuous random variable X. Suppose that
the sample ($X_1,..., X_n$) is not a random sample, but for $n \ge 2$, the joint pdf of the
order statistics ($X_{(1)},..., X_{(n)}$) can be written as
$f_{X_{(1)},...,X_{(n)}}(x_1,...,x_n) = f(x_n) \prod_{i=1}^{n-1} f(x_i)/(1 - F(x_i))$, $x_1 < x_2 < ... < x_n$,
and 0 otherwise.
(a) Suppose that
$f(x) = \begin{cases} exp(-x) & x > 0, \
0 & \text{otherwise}, \end{cases}$
and that $W_1 = X_{(1)}$ and $W_i = X_{(i)} - X_{(i-1)}$ for $i = 2,...,n$. Show that
$W_1, W_2,..., W_n$ are mutually independent.
February 22, 2024
(b) Determine the exact distribution of $X_{(n)}$ using the result in (a).
(c) Show that $X_{(n)}$ is asymptotically normal. Identify the mean and variance of the
normal distribution.
