Engineering Probability 2018 Exam 2 - Thu 2018-03-29
Name, RCSID:
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Rules:
- You have 80 minutes.
- You may bring two 2-sided 8.5"x11" paper 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 eleven pages.
- Each part of a question is worth 5 points.
- You may cross out two questions, which will not be graded.
- When answering a question, don't just state your answer, prove it.
Questions:
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Consider this probability distribution:
$$f_X(x)= \begin{cases} a(2-x) & \text{if } 0\le x\le1\\ 0&\text{otherwise}\end{cases}$$
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What is $a$?
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What is $F_X(x)$?
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What is E[X]?
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What is the reliability, R[x]?
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What is the MTTF?
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What is the failure rate, r(x)?
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What is $f_X(x|x>.5)$?
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Define a new r.v. Y=2X, where X is the r.v. in the previous question.
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What is $f_Y(y)$?
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What is $F_Y(y)$?
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What is E[Y]?
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Your web server gets on the average 1 hit per second. The possible clients are independent of each other.
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What is the name of appropriate distribution for the number of hits per second?
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What is the probability that it gets exactly one hit in the next two seconds?
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What is the name of appropriate probability distribution for the time between successive hits?
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What is the probability that the time between two successive hits is less than two seconds?
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Let X be an exponential random variable with mean 1.
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Using the Markov inequality, what's P[X>3]?
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Using the Chebyshev inequality, what's P[X>3]?
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What's the exact P[X>3]?
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Let X be a normal random variable with mean 100 and standard deviation 10. Give the following numbers, using the supplied table.
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P[X>100].
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P[80<X<100].
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You're tossing 10000 fair coins. What's the probability of getting between 5000 and 5100 heads? Use the table.
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Evaluate $$\int_0^\infty e^{-2 x^2} dx$$
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Let $f_X(x) = 1$ and $f_Y(y)=2y$, both in the range $0\le x, y\le1$.
Let Z=max(X,Y).
What is E[Z]?
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Normal distribution:
x f(x) F(x) Q(x) -3.0000 0.0044 0.0013 0.9987 -2.9000 0.0060 0.0019 0.9981 -2.8000 0.0079 0.0026 0.9974 -2.7000 0.0104 0.0035 0.9965 -2.6000 0.0136 0.0047 0.9953 -2.5000 0.0175 0.0062 0.9938 -2.4000 0.0224 0.0082 0.9918 -2.3000 0.0283 0.0107 0.9893 -2.2000 0.0355 0.0139 0.9861 -2.1000 0.0440 0.0179 0.9821 -2.0000 0.0540 0.0228 0.9772 -1.9000 0.0656 0.0287 0.9713 -1.8000 0.0790 0.0359 0.9641 -1.7000 0.0940 0.0446 0.9554 -1.6000 0.1109 0.0548 0.9452 -1.5000 0.1295 0.0668 0.9332 -1.4000 0.1497 0.0808 0.9192 -1.3000 0.1714 0.0968 0.9032 -1.2000 0.1942 0.1151 0.8849 -1.1000 0.2179 0.1357 0.8643 -1.0000 0.2420 0.1587 0.8413 -0.9000 0.2661 0.1841 0.8159 -0.8000 0.2897 0.2119 0.7881 -0.7000 0.3123 0.2420 0.7580 -0.6000 0.3332 0.2743 0.7257 -0.5000 0.3521 0.3085 0.6915 -0.4000 0.3683 0.3446 0.6554 -0.3000 0.3814 0.3821 0.6179 -0.2000 0.3910 0.4207 0.5793 -0.1000 0.3970 0.4602 0.5398
Normal distribution:
x f(x) F(x) Q(x)
0 0.3989 0.5000 0.5000
0.1000 0.3970 0.5398 0.4602
0.2000 0.3910 0.5793 0.4207
0.3000 0.3814 0.6179 0.3821
0.4000 0.3683 0.6554 0.3446
0.5000 0.3521 0.6915 0.3085
0.6000 0.3332 0.7257 0.2743
0.7000 0.3123 0.7580 0.2420
0.8000 0.2897 0.7881 0.2119
0.9000 0.2661 0.8159 0.1841
1.0000 0.2420 0.8413 0.1587
1.1000 0.2179 0.8643 0.1357
1.2000 0.1942 0.8849 0.1151
1.3000 0.1714 0.9032 0.0968
1.4000 0.1497 0.9192 0.0808
1.5000 0.1295 0.9332 0.0668
1.6000 0.1109 0.9452 0.0548
1.7000 0.0940 0.9554 0.0446
1.8000 0.0790 0.9641 0.0359
1.9000 0.0656 0.9713 0.0287
2.0000 0.0540 0.9772 0.0228
2.1000 0.0440 0.9821 0.0179
2.2000 0.0355 0.9861 0.0139
2.3000 0.0283 0.9893 0.0107
2.4000 0.0224 0.9918 0.0082
2.5000 0.0175 0.9938 0.0062
2.6000 0.0136 0.9953 0.0047
2.7000 0.0104 0.9965 0.0035
2.8000 0.0079 0.9974 0.0026
2.9000 0.0060 0.9981 0.0019
3.0000 0.0044 0.9987 0.0013
End of exam 1, total 100 points (considering that 2 questions aren't graded).