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ECE 603 - Probability and Random Processes, Fall 2006
Final Exam
December 15th, 10:30am-12:30pm, Marston 132
Overview
The exam consists of five problems for 120 points. The points for each part of each problem are given in brackets - you should spend your two hours accordingly. The exam is closed book, but you are allowed three page-sides of notes. Calculators are not allowed. I will provide all necessary blank paper.
Testmanship
Full credit will be given only to fully justified answers.
Giving the steps along the way to the answer will not only earn full credit but also maximize the partial credit should you stumble or get stuck. If you get stuck, attempt to neatly define your approach to the problem and why you are stuck. If part of a problem depends on a previous part that you are unable to solve, explain the method for doing the current part, and, if possible, give the answer in terms of the quantities of the previous part that you are unable to obtain. Start each problem on a new page. Not only will this facilitate grading but also make it easier for you to jump back and forth between problems. If you get to the end of the problem and realize that your answer must be wrong, be sure to write โthis must be wrong because... โ so that I will know you recognized such a fact. Academic dishonesty will be dealt with harshly - the minimum penalty will be an โFโ for the course.
Hint: You may find the following fact useful as you solve this exam:
Fact: We know from class that any linear combination
of jointly Gaussian random variables (^) (^) is Gaussian. The converse is also true: If all linear combinations are Gaussian,
then the random variables ^ ^ are jointly Gaussian.
- I am running a machine that produces parts. The machine has good days and bad days. On a good day, each part produced has a 10% chance of being defective; on a bad day, each part produced has a 50% chance of being defective. Given the type of day , the event of a part being defective is independent of the event of any other part being defective on that day. The machine has good days 75% of the time and bad days 25% of the time. Give expressions for the following quantities:
[5] (a) Assuming that today is a good day, find the probability that there are 5 defective parts in a batch of 20 parts produced by the machine today.
[5] (b) Without knowledge of whether today will be a good day or a bad day, find the probability that there are 5 defective parts in a batch of 20 parts produced by the machine today.
[7] (c) Suppose I test the first 5 parts made by the machine on a given day and find that 3 of the parts are defective. Find the probability that this is a good day and the probability that this is a bad day.
[8] (d) Suppose I test the first 5 parts made by the machine on a given day and find that 3 of the parts are defective. Find the probability that there will be 5 defective parts in the first 20 parts made that day (including the 5 parts that you have tested).
- The random variables and have joint probability density function
6 < (^) otherwise
[5] (a) Find the value of
.
[10] (b) Find
>= !$#^ :%?'
, the conditional probability density function of given @)A%^. Sketch = !$# (^) : 6 <CBD' (i.e. the pdf for given that E)F6<CB^ ).
[5] (c) Write an expression (no need to evaluate) for G
IH41"/^ '.
[10] (d) Define the function J
LK '
as:
J
!$# 'M)
HN
6 < (^) else
Let O ) J
'. Find
DP*! Q '
, the probability density function of O.
