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Final exam, Stat/Elec 331, Fall 2003
Solutions should be clear, complete and easy to follow. You are allowed to use the book, a calculator, your lecture notes, and lecture notes posted on the course web page. The time limit is five hours. Each problem is worth 6 points. The deadline to turn in the exam is December 17, 5 pm. Late turn-ins are not accepted.
- The continuous random variable X has pdf
f (x) =
{ 1 / 2 if 0 < x โค 1 1 /(2x^2 ) if x โฅ 1
a. Show that this is a possible pdf for a continuous random variable.
b. Compute E[X].
c. Find the cdf of X and sketch its graph.
d. Let Y = 1/X. Find the cdf of Y and sketch its graph.
- A company manufactures metal plates of size 5 x 10 (inches). Due to random fluctuations, a manufactured plate has a size of X x Y inches where (X, Y ) follows a bivariate normal distribution with means 5 and 10, variances 0.01 and 0.04 and correlation coefficient 0.8. Let C be the circumference (perimeter) and A the area of a plate
a. Find E[C] and E[C|X = x].
b. Find E[A] and E[A|X = x].
c. A plate is useful if 29 โค C โค 31. What is the probability that a plate is useful?
d. Now suppose that you have ten plates. What is the probability that at least nine are useful?
- The random variable X has pdf
f (x) = (a + 1)xa, 0 โค x โค 1
where a is an unknown parameter.
a. Find the maximum likelihood estimator and the moment estimator of a based on a sample X 1 , X 2 , ..., Xn.
b. Suppose a = 2. Describe how you can simulate observations on X based on observations from a uniform [0,1]-distribution. If such a uniform value is 0.008, what value of X does this give?
- Accidents on a certain road occur according to a Poisson process with rate ฮป accidents/week.
a. The two towing companies A and B have agreed to take turns in dealing with the accidents. Thus, A takes care of the first accident, B the second and so on. Consider the process of accidents that A takes care of. Is this a Poisson process? If so, what is the rate?
b. Suppose that in a particular year, it is observed that N of the 52 weeks had no accidents. What is the distribution of N (name and parameters)?
c. Based on N, find the moment estimator of ฮป. Note that we only have one observation on N, so our sample size is n = 1.
(a) (b)
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โ
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2
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โ
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0
1
(c) (d)
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2
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4
โ4 โ4 โ3 โ2 โ1 0 1 2 3 4
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(e) (f)
โ50 โ4 โ3 โ2 โ1 0 1 2 3 4
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10
20
30
40
50
โ4 โ4 โ3 โ2 โ1 0 1 2 3 4
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(g) (h)
โ10 โ10 โ8 โ6 โ4 โ2 0 2 4 6 8 10
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โ4 โ4 โ3 โ2 โ1 0 1 2 3 4
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