Assignment 3

[This assignment covers the work in Distribution Theory Chapters 5 and 6.]

1. Let $X_1$, $X_2$, $X_3$ be a random sample from a distribution with pdf
   
   $$f(x) = 2x, 0 \leq x \leq 1.$$
   
   
   Find
   1. the pdf of the smallest of these, $Y_1$;
   2. the probability that $Y_1$ exceeds the median of the distribution.
2. Let $Y_1< Y_2 < Y_3 < Y_4$ be an ordered sample from a distribution with pdf $f(x) = \alpha e^{- \alpha x}$, $x \geq 0$. Find
   1. the distribution of $Y_4$;
   2. $P(Y_4<1)$;
   3. the joint distribution of $Y_1$ and $Y_3$.
3. $X_1$, $X_2$, ..., $X_n$ is a random sample from a continuous distribution with pdf f(x). An additional observation, $X_{n+1}$, say, is taken from this distribution. Find the probability that $X_{n+1}$ exceeds the largest of the other n observations.
4. Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5$ denote the ordered sample from a distribution having pdf $f(x)=3x^2$, $0<x<1$. Show that $Z_1=Y_2/Y_4$ and $Z_2=Y_4$ are stochastically independent.
5. Three independent samples, each of size n are drawn from a U(0,1) distribution. Let $Z_1$, $Z_2$, $Z_3$ denote the largest order statistics in the 3 samples respectively, and let $U=Z_1Z_2Z_3$.
   1. Find the pdf of U.
   2. Generalize this result for m independent samples.
6. Let $Y_m$ and $Z_n$ be the largest order statistics from 2 independent samples of sizes m and n respectively from a U(0, 1/$\alpha$) distribution. Let $U=Y_m/Z_n$.
   1. Find the pdf of U.
   2. Find $P(U \geq c)$ where $c > 0$.
   3. Explain how this can be used to test the hypothesis that 2 samples of size m and n have come from the same uniform distribution.
7. Let $F(Y_i)-F(Y_{i-1})=W_i$. Note that $W_i$ is the area under the graph of the pdf $f(x)$ between $x=Y_{i-1}$ and $x=Y_i$. Then $W_i$ is called a coverage of the random interval $\{x: \, Y_{i-1}<x<Y_i\}$. In the notation of Theorem 4.3,
   
   $$W_i=Z_i-Z_{i-1}, \ i=1, 2, \ldots, n+1,$$
   
   
   where we need to define $Z_0=0$ and $Z_{n+1}=1$.
   1. Find the distribution of $W_i$.
   2. Show that $E(W_i)=1/(n+1)$ for all $i$.
   3. Show that the joint pdf of the n coverages $W_1, W_2$, ..., $W_n$ is
      
      $$f_{W_1, \ldots, W_n}(w_1, \ldots, w_n)=n!, \ w_i>0,\ i=1, \ldots, n,\ \sum_{i-1}^nw_i < 1.$$
      
      

   4. Find $E(F(Y_j)-F(Y_i))$, where $i<j$.
8. Derive the mean and variance of a random variable W $\sim \chi_p^2(\lambda)$.