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# Assignment 3

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

1. Let , , be a random sample from a distribution with pdf

Find
1. the pdf of the smallest of these, ;
2. the probability that exceeds the median of the distribution.
2. Let be an ordered sample from a distribution with pdf , . Find
1. the distribution of ;
2. ;
3. the joint distribution of and .
3. , , ..., is a random sample from a continuous distribution with pdf f(x). An additional observation, , say, is taken from this distribution. Find the probability that exceeds the largest of the other n observations.
4. Let denote the ordered sample from a distribution having pdf , . Show that and are stochastically independent.
5. Three independent samples, each of size n are drawn from a U(0,1) distribution. Let , , denote the largest order statistics in the 3 samples respectively, and let .
1. Find the pdf of U.
2. Generalize this result for m independent samples.
6. Let and be the largest order statistics from 2 independent samples of sizes m and n respectively from a U(0, 1/) distribution. Let .
1. Find the pdf of U.
2. Find where .
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 . Note that is the area under the graph of the pdf between and . Then is called a coverage of the random interval . In the notation of Theorem 4.3,

where we need to define and .
1. Find the distribution of .
2. Show that for all .
3. Show that the joint pdf of the n coverages , ..., is

4. Find , where .
8. Derive the mean and variance of a random variable W .

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Bob Murison 2000-10-31