Engineering Probability Exam 3  Sat 20190504
Name, RPI email:
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Rules:
 You have 80 minutes.
 You may bring three 2sided 8.5"x11" papers with notes.
 You may bring a calculator.
 You may not share material with each other during the exam.
 No collaboration or communication (except with the staff) is allowed.
 Check that your copy of this test has all nine pages.
 Each part of a question is worth 5 points.
 When answering a question, don't just state your answer, prove it.
 You may write FREE as your answer for two questions, and get the 5 points.
Questions:

These few questions are about the population of adult males, which has a mean of 70 inches and a standard deviation of 4 inches.

What is the probability that a particular person's height is between 68 and 74?
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If we take a sample of 100, what is its mean?
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What is its standard deviation?
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These questions are about tossing 3 fair dice and looking at the 3 numbers that show. However, these dice have only 2 faces (to make this question easier). The faces are labeled 1 and 2.

What's the expected value of the number showing on the first die?
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What's the pmf of the smallest die?
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What's the expected value of the smallest die?
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What's the probability that the smallest number is 1 given that the first number is 2?
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What's the probability that the first number is 2 given that the smallest number is 1?
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Are the first number and the smallest number are independent? Prove your answer.
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What is the MAP estimator for the smallest number, given that the first number is 2?
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This question is about a continuous probability distribution on 2 variables.
$$f_{XY}(x,y) = \begin{cases} c (x+y) & \text{ if } (0\le x) \ \& \ (0\le y)\ \& \ (0\le x+y \le 1) \\ 0 & \text{ otherwise}\end{cases}$$
The nonzero region is the triangle with vertices (0,0), (1,0) and (0,1).
c is some constant, but I didn't tell you what it is.

What is c?
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What is $F_{XY}(x,y)$?
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What is $f_X(x)$?
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Are X and Y independent?
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What is $P[X\le Y]$ ?
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What is $E[X]$?
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What is $COV[X,Y]$?
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What is $\rho_{X,Y}$?
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What is $f_Y(yx)$?
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What is $E[Yx]$?
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I'm comparing two types of widgets, red and blue. Assume that the probability of each widget dieing in a small interval dt, given that it was alive at the start, is independent of its age. Assume that the probability of the red widget dieing in the next hour is 0.1%, for the blue, it's 0.01%.

Give the pdf for the red widget's lifetime. (You have enough info to do this; there is only one possible probability distribution.)
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If you have 100 red widgets, what's the probability that their mean lifetime is within 10% of the population mean?
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If you start two widgets at the same time, what's the probability that the red widget will last longer?
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Normal distribution:
x f(x) F(x) Q(x) x f(x) F(x) Q(x) 3.0000 0.0044 0.0013 0.9987 0.1000 0.3970 0.5398 0.4602 2.9000 0.0060 0.0019 0.9981 0.2000 0.3910 0.5793 0.4207 2.8000 0.0079 0.0026 0.9974 0.3000 0.3814 0.6179 0.3821 2.7000 0.0104 0.0035 0.9965 0.4000 0.3683 0.6554 0.3446 2.6000 0.0136 0.0047 0.9953 0.5000 0.3521 0.6915 0.3085 2.5000 0.0175 0.0062 0.9938 0.6000 0.3332 0.7257 0.2743 2.4000 0.0224 0.0082 0.9918 0.7000 0.3123 0.7580 0.2420 2.3000 0.0283 0.0107 0.9893 0.8000 0.2897 0.7881 0.2119 2.2000 0.0355 0.0139 0.9861 0.9000 0.2661 0.8159 0.1841 2.1000 0.0440 0.0179 0.9821 1.0000 0.2420 0.8413 0.1587 2.0000 0.0540 0.0228 0.9772 1.1000 0.2179 0.8643 0.1357 1.9000 0.0656 0.0287 0.9713 1.2000 0.1942 0.8849 0.1151 1.8000 0.0790 0.0359 0.9641 1.3000 0.1714 0.9032 0.0968 1.7000 0.0940 0.0446 0.9554 1.4000 0.1497 0.9192 0.0808 1.6000 0.1109 0.0548 0.9452 1.5000 0.1295 0.9332 0.0668 1.5000 0.1295 0.0668 0.9332 1.6000 0.1109 0.9452 0.0548 1.4000 0.1497 0.0808 0.9192 1.7000 0.0940 0.9554 0.0446 1.3000 0.1714 0.0968 0.9032 1.8000 0.0790 0.9641 0.0359 1.2000 0.1942 0.1151 0.8849 1.9000 0.0656 0.9713 0.0287 1.1000 0.2179 0.1357 0.8643 2.0000 0.0540 0.9772 0.0228 1.0000 0.2420 0.1587 0.8413 2.1000 0.0440 0.9821 0.0179 0.9000 0.2661 0.1841 0.8159 2.2000 0.0355 0.9861 0.0139 0.8000 0.2897 0.2119 0.7881 2.3000 0.0283 0.9893 0.0107 0.7000 0.3123 0.2420 0.7580 2.4000 0.0224 0.9918 0.0082 0.6000 0.3332 0.2743 0.7257 2.5000 0.0175 0.9938 0.0062 0.5000 0.3521 0.3085 0.6915 2.6000 0.0136 0.9953 0.0047 0.4000 0.3683 0.3446 0.6554 2.7000 0.0104 0.9965 0.0035 0.3000 0.3814 0.3821 0.6179 2.8000 0.0079 0.9974 0.0026 0.2000 0.3910 0.4207 0.5793 2.9000 0.0060 0.9981 0.0019 0.1000 0.3970 0.4602 0.5398 3.0000 0.0044 0.9987 0.0013 0 0.3989 0.5000 0.5000
End of exam 3, total 100 points.