First Exam

Name, RCSID:

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Rules:

1. You have 80 minutes.
2. You may bring one 2-sided 8.5"x11" paper with notes.
3. You may bring a calculator.
4. You may not share material with each other during the exam.
5. No collaboration or communication (except with the staff) is allowed.
6. Check that your copy of this test has all six pages.
7. You may omit questions totalling 4 points. You must cross out the ones you omit, or we will assume that you omitted the last subquestion.
8. When answering a question, don't just state your answer, prove it.

Questions:

1. This is a question about smoking and lung cancer. C is the event that someone has cancer. S is the event that someone smokes. Assume that

   1. 10% of smokers get lung cancer. P(C|S)=.1
   2. 90% of lung cancers happen to smokers. P(S|C)=.9
   3. Assume that 20% of people smoke. P(S)=.2

   Questions:

   1. (2 points) What is P(C)?

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   2. (2 points) What is P(C|S')?

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2. You are trying to pass the very difficult course ECSE-3030. For each time you try, you pass with probability 1/2. Your chance of passing this time is independent of how many times you've tried.

   The random variable is the number of times you have to take the course until you pass for the first time. E.g., if you pass on the first time, this number is one.

   1. (2 pts) What's the relevant probability distribution?

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   2. (2 pts) What's the expected number of times you will have to take the course?

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   3. (2 pts) What's the standard deviation?

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3. This question is about tossing a 20-sided fair die, faces labeled from 1 to 20.

   1. Event A is that a number up to 10 shows.
   2. Event B is that an odd number shows.
   3. Event C is that the number is in the set {2, 4, 6, 8, 10, 11, 13, 15, 17, 19}.

   Questions:

   1. (2 points) Are A and B independent? Don't just say, yes or no. Prove your answer.

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   2. (2 points) Are A and C independent?

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   3. (2 points) Are B and C independent?

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   4. (4 points) Are A, B, and C independent?

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4. This question is about transmitting a signal over a noisy channel. The source transmits either 0 or 1. However, you receive one of three (not two) signals: A, B, or C.

   1. P(0)=.2, P(1)=.8
   2. P(A|0)=.8, P(B|0)=P(C|0)=.1
   3. P(A|1)=P(B|1)=.2, P(C|1)=.6

   Questions:

   1. (6 points) What are P(A&0), P(B&0), P(C&0)?

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   2. (6 points) What are P(A&1), P(B&1), P(C&1)?

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   3. (6 points) What are P(A), P(B), P(C)?

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   4. (6 points) What are P(0|A), P(0|B), P(0|C)?

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5. An LCD display has 2000 * 2000 pixels. A display is accepted if it has 10 or fewer faulty pixels. The probability that a pixel is faulty coming out of the production line is 1e-6.

   1. (2 points) What's the appropriate probability distribution for the number of bad pixels in a display?

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   2. (2 pts) What's the mean number of bad pixels in a display?

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   3. (2 pts) What's the probability that a display has all good pixels?

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   4. (4 pts) What proportion of displays are accepted? An expression is ok; you don't need the actual number.

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End of exam 1, total 50 points.