Exercise 4 (11 points)
Assume that $X_1, X_2, X_3, \dots, X_n$ are iid continuous random variables with density $f$ and
distribution function $F$.
1) Show that the density of the minimum is given by
$f_{(1)}(x) = nf(x)(1 - F(x))^{n-1}$.
2) Assume $X_1, X_2, X_3, \dots, X_n \sim Exp(\lambda)$. Find the distribution of
(a) $X_{(1)}$
(b) The median.
3) Assume $X_1, X_2, X_3, \dots, X_n \sim Unif(0, 1)$. Find the distribution of $X_{(1)}$ (specify the
parameters).
(5 points)
(2 points)
(2 points)
(2 points)
