STAT 801

Problems: Assignment 3

Postscript version of these questions

1.

Suppose $X_1, \ldots , X_n$ are iid real random variables with density f . Let $X_{(1)},\ldots,X_{(n)}$ be the X's arranged in increasing order.

(a)

Find the joint density of $X_{(1)},\ldots,X_{(n)}$ .

(b)

Suppose f=1_[0,1]. Prove that $(X_{(1)}/X_{(k)}, \ldots , X_{(k-1)}/X_{(k)})$ is independent of $(X_{(k)}, \ldots ,X_{(n)})$.

(c)

Find the density of X_(k).

(d)

Find the density of X_(k)-X_(j).

2.

Suppose $X_1,\ldots ,X_{n+1}$ are iid exponential. Let $S_m = \sum_1^m X_i$.

(a)

Find the joint density of $(X_1 /S_{n+1}, \ldots ,X_n/S_{n+1})$.

(b)

Find the joint density of $(S_1 /S_{n+1}, \ldots ,S_n /S_{n+1})$.

3.

Suppose $X_1, \ldots , X_n$ are iid N($\mu$,$\sigma^2$). Let ${\bar X }_m = (X_1+ \cdots + X_m)/m$. Let $S_m^2 = \sum_1^m (X_i-{\bar X}_m ) ^2$.

(a)

Develop a recurrence relation for S_m and ${\bar X}_m$, expressing S_m and ${\bar X}_m$ in terms of S_m-1 and ${\bar X}_{m-1}$.

(b)

Find the joint density of $({\bar X}_n ,S_2^2, \ldots ,S_n^2)$.

(c)

Generate data from N(0,1). By adding 10^k to the data for some large values of k compare the numerical performance of these recurrence relations to that of the one pass formula using $T_1=\sum_1^nX_i$, $T_2 = \sum X_i^2$ and the usual computing formulas for the sample variance.

4.

Suppose X and Y are iid $N(0, \sigma^2)$.

(a)

Show that X²+Y² and X/(X²+Y²)^1/2 are independent.

(b)

Show that $\Theta = \arcsin(X/(X^2 +Y^2)^{1/2})$ is uniformly distributed on $(-\pi/2,\pi/2]$.

(c)

Show X/Y is a Cauchy random variable.