ECSE-2500 Engineering Probability, Exam 2, April 5, 2011









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There are 5 pages with 4 questions. Answer any 3 questions. (If you answer 4, then we will grade the first 3.)

We record the order in which you hand in your exam the better to keep track of the exams, and sometimes to plot scatterplots of order vs grade. (There is never any significant correlation.) Your order does not affect your grade.

This exam is closed book but a calculator and two 2-sided letter-paper-size note sheets are allowed. You may not use other computers or communication devices, or share material with other students.

1. In your body, an average of 4400 radioactive potassium atoms decay each second. (ref: http://en.wikipedia.org/wiki/Becquerel). You're interested in the number of decays in one millisecond.
1. [_____/1] What's the relevant probability distribution?

2. [_____/1] What's the value of the relevant parameter?

3. [_____/2] What are the mean and standard deviation of the number of decays in one millisecond?

4. [_____/2] What's the probability that there will be zero decays? Write this as a formula. You don't need to give a number.

5. [_____/1] What's the relevant probability distribution for the time between consecutive decays?

6. [_____/1] What's the mean time between consecutive decays?

7. [_____/1] What's the standard deviation of the time between consecutive decays?

8. [_____/1] What's the probability that the time between consecutive decays is greater than one 1msec?

2. I write a digit that is either a 0 or a 1. You have trouble reading my writing. This table shows the joint probability of what I write and you read. X is the random variable for what I write, Y the random variable for what you read.
 X=0 X=1 Y=0 .4 .2 Y=1 .1 .3
1. [_____/1] What's the marginal pmf of X?

2. [_____/1] What's the mean of X?

3. [_____/1] What's the variance of X?

4. [_____/2] What's the mean of XY?

5. [_____/2] What's the covariance of X and Y?

6. [_____/2] What's the correlation coefficient of X and Y?

7. [_____/1] If you read a 0, what's the probability that you're wrong?

3. [_____/10] Evaluate and give me the answer as an expression perhaps involving {$\pi$} and {$\sqrt{\cdot}$}. Show your work. {$${\cal A} = \int_{0}^\infty e^{-x^2} dx$$} Hint: Transform this to something closer to {$$\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}} dx =1$$} .

4. You're bicycling the Alaska Highway. Assume that the lifetime of a bicycle tire is exponential with mean {$\mu=3$} days. You carry 3 spares.
1. [_____/3] What is the expected number of days before you do not have 2 good tires to bicycle on? (Hint: You do not need to use any distribution that we didn't study. In fact, there is a relatively easy way to compute the answer.)

2. [_____/1] Let's assume that the answer is a mean of 5 days. (It is not.) Using only that info, and nothing else about the distribution, give an upper bound on the probability that you're still bicycling after 10 days.

3. [_____/2] Let's further assume that the answer to the first part has a standard deviation of 5 days. (It does not.) Using only that mean and standard deviation info, and nothing else about the distribution, give an upper bound on the probability that you're still bicycling after 10 days.

4. [_____/1] Assume that your front tire is two days old, and is still good. What's its expected future lifetime?

5. [_____/3] What's the probability that both tires will fail in the first hour? It's ok to write an expression with an exponential. It's ok to ignore the probability that a replacement tire will also fail since that's low. Assume that the tires are independent (though in the real world their failures could be correlated.)

(end of exam)