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# Homework Assignments

## Homework 4

1. Suppose .
1. Show that and .
2. Show that .
3. Suppose and write . Show that .
2. Suppose you have a random number generator which is capable of generating random numbers distributed as . Describe how to generate random vectors .

3. Suppose and are r.v.s. Show that is minimized over all functions when is the function

Assume .

4. Let . Let , where

Determine the relationship between and and .

5. Suppose where

1. The value is measured. Determine the best estimate for .
2. In a separate problem, the values and are measured. Determine the best estimate of .
3. Determine a random vector which is a whitened version of .
Problems from Grimmet & Stirzaker
1. Ex 3.7.5. What is requested is , i.e., the mean subsequent lifetime given that the machine is still running after days. Then use the hint from the book. Note that in (a), .
2. Ex 3.7.7. Hint: Show that robot faulty fault not detected . Hence argue that the number of faulty passed robots, given , is distributed as , which has mean . Hence show that .
3. Ex 4.1.1(a)
4. Ex 4.1.2
5. Ex 4.2.1. Hint: think geometric r.v.
6. Ex 4.2.2. Hint: max .
7. Ex 4.4.1. Hint: integrate by parts.
8. Ex 4.6.4(b)