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Unformatted text preview: C)" ECE 302, Midterm #2
7:00—8:0013111 Thursday, Feb. 22, BB 170, . Enter your name, student ID number, e—mail address, and signature in the space provided on this page, NOW! . This is a closed book exam. . This exam contains 5 questions. You have one hour to complete it. I will suggest not spending too much time on a single question, and work on those you know how
to solve. . The sub—questions of a given question are listed from the easiest to the hardest. The best strategy may be to finish only the sub—questions you know exactly how to
solve. . There are a total of 11 pages in the exam booklet. Use the back of each page for rough work. . Neither calculators nor help sheets are allowed. Read through all of the problems ﬁrst, and consult with the TA during the ﬁrst
15 minutes. After that, no questions should be asked unless under special circum
stances, which is at TA’s discretion. You can also get a feel for how long each
question might take after browsing through the entire question set. Good luck! Name:
Student ID:
E—rnail: Signature: Question I: [2000] A salesperson travelled between cities A, B, and C for four consecutive
nights, and each night he could only stay in one city. Suppose the populations of cities
A, B, and C' are 2, 1, and 0.5 millions respectively At day 1, the salesperson started from city A and stayed there for the ﬁrst night. For
the next three mornings, the salesperson randomly selected the next destination from the
two cities excluding the city he stayed for the last night, and the probability he chose a
certain city was proportional to the population of that city. For example, if he stayed
in city B for the third night, then he could only spend the fourth night in either city A
or city C. And the probabilities that he chose cities A and C are P(AlB) = c X 2 and
P(CB) = c X 0.5 for a common 0. 1. [5%] What is the sample space?
2. Use the tree method to specify the weight assignment for the sample space.
3. What is the probability that the salesperson was able to visit all three cities? 4. [5%] What is the probability that the salesperson visited city B twice, given that
the salesperson had visited all three cities? 1 / l ‘
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a, A 53; r: A C; Question 2: [20%] Consider a continuous random variable X with its pdl' as follows. +1) if :1: E [~l,0] = c(1— 11:) if :1; E (0, 1] 0 otherwise.
1. [5%] Find the value of c. 2, [5%] What is the deﬁnition of the cumulative distribution function (usually denoted
by EH33»? 3. [5%] Find the cdf 4. [5%] Consider two events: A = {X g 0} and B = {X2 > Are A and B independent? Question 3: {20%} Consider two independent random variables X and Y. X is a continuous
uniform random variable distributed between (~2, 2). Y is a discrete random variable with
equal probability weight among {—1, 1}. 1.
2. [5%] Find the pdf fy(y). Express it as a single equation and plot fy(y). [5%] Suppose Z = X+Y. Find the cdf 172(2). Hint: P(Z S 2) = P(Z S 2 and X =
—1)+ P(Z E z and X =1). [5%] Find the conditional probability P(X = —1lZ < 0.5). [5%] Find the pdf fZ(z). If you do not know the answer to the above question, then
you may assume the 172(2) is as follows and solve the question accordingly. (Note
this is an incorrect assumption.) 0 if33<0
172(2): 0.5x ifOSac<1
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r Question 4: [20%] A factory is producing GPS chips. From the historic data, there are
in average 20 defective chips per 24 hours being produced. 1. [10%] A consumer electronics company wants to place a big order of GPS chips
and will pay this factory a Visit. During the 3—hour Visit, if there are more than
2 defective chips being produced, the company will cancel the order. What is the
probability that the order will be cancel? You can assume the number of defective
chips is Poisson distributed. 2. [10%] Find out the conditional probability that
P(there are precisely 2 defective chipslthe order is cancelled). But this time, we assume the number of defective chips is geometrically distributed
with parameter p = 0.9. eases... 5" m
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‘13; or; it”: ﬁts ,. I ‘ i Question 5: [2500] Consider X being the snow precipitation of February (with unit:
“inch”). From the historical data (or we may assume that), X is a Gaussian distribution
with m = 20, a = 10. Your answers have to be computed/ converted to numbers or ratio—
nal numbers “(z/b” instead of only writing down the integrals. You may want to use the
included table of <I>(a;), the cumulative distribution of a standard Gaussian distribution. 1. What is the deﬁnition of the standard Gaussian distribution?
2. [5%] Find the probability that P(X g 20). 3. [5%] Find the probability that P(20 g X < 40). 4. [5%] Find the probability that P(5 S X < 10). 5. [5%] Conditioning that there have already been 25 inches of snow between 2 / 1—2 / 21, What is the probability that there will be an additional 10 inches of snow in the next
week (between 2 / 22w2 / 28). @(0.5) = 0691, 0(1) = 0.841, <I>(1.5) = 0.9332, 9(2) = 0.9662, <I>(2.5) = 0.99379,
@(3) = 0.99865. as»; 3:,» “2g E M3 5 w 4‘ A
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 Spring '08
 GELFAND
 Probability, Probability distribution, Probability theory, CDF

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