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Unformatted text preview: NTUEE Probability Ba Statistics 2006 Spring. Probability Midterm Exam 2:204:20 pm Thursday. 20 April 2006
Total: 3 pages. 100 points. (i) A calculator is NOTaZioweri. You may not eoasaft‘ the textbook or neighbor: (2)Please rare oﬂyoar cellphones before the exam.
(3)Piease show your workfor pariiai credit. and underline your answers. Points are awarded for solutions. not answers. so correct answers without justiﬁcation will not receive fall
credit. PROBLEM 1: (8%)
Deﬁne a function of two events A and B by i. P AUB
s(A;B) =—~——*( )
PCB)
For all events A. B. with P(A) denoting the probability of an event A. Now. for any given event B. can g(A.‘B) be a probability measure? (Hint: A probability measure MA) has to satisfy the Three Axioms of Probability. Namely. (1)
MA) 2 0 for any eventA. (2) MS) 2 1 for the sample space S. and (3) If A F) C is an empty set
for any events A and C. then P(A U C) : PTA) + P(C).) PROBLEM 2: (6%)
Consider rolling three fair dice (of the same size) at a time in a casino. The game is to bet on the numbers that the three dice show up. Let n1. ng. and n3 be the three numbers that show up
in a single roll. (a) Find the probability of the event that r11 = 112 = r13. (2%) (b) Find the probability of the event that 111 <1 n; *1: n3. (2%) (0) Find the probability of the event that n; + n; + 113 = 12. (2%) PROBLEM 3: (6%) For events A. B. C. solve the following questions: ($119514): 0.2. P(A U B) = 0.5. and P(A[B) = 0.5. ﬁnd P(B). (2%) (b) If and B are mutually exclusive and independent and if B C A. find P(B). (2%)
(e) Show that 2P(A t) B) E P(A) + 13(8). (2%) PROBLEM 4: (5%)
Prove that the inequality P[0A.]s;P(A.) k=l holds for arbitrary events A .r. A 3. A... with ii a positive integer. GO ON TO THE NEXT PAGE. NTUEE Probability & Statistics 2006 Spring PROBLEM 5: (7%) A communication channel transmits the digits 0 and 1. However, due to static, the digit
transmitted is incorrectly received with probability 0.2. Suppose that we want to transmit an
important message consisting of one binary digit. To reduce the chance of error, we transmit
00000 instead of 0 and Illll instead of i. If the receiver of the message uses “majority”
decoding, what is the probability that the message will be incorrectly decoded? What
independence assumptions are you making? (By majority decoding we mean that the message is decoded as “0” if there are at least three zeros in the message received and as “1” otherwise.) PROBLEM 6: (9%)
According to a representative for an automobile manufacturer, the company uses 3000 iock—and—key combinations on its vehicles. Suppose that you ﬁnd a key for one of these cars. (a) Give the expected number of vehicles that you would have to check to ﬁnd one that your
key ﬁt. (3%) (b) Give the probability that you would have to check at least 3000 vehicles to ﬁnd one that
your key ﬁt. (3%) (c) Give the probability that at most 2000 vehicles would have to be checked to find one that
your key ﬁt. (3%) (Note. Please give simplest form of your answers as you can.) PROBLEM 7: (9%)
The number of times that an individual contracts a cold in a given year is a Poisson random variable with parameter k = 3, Suppose a new wonder drug (based on large quantities of
vitamin C) has just been marketed that reduces that Poisson parameter to 9x. = 2 for 75 percent of the population. For the other 25 percent of the population; the drug has no appreciable
effect on colds. If an individual tries the drug for a year and has 0 colds in that time, how likeiy is it that the drug is beneficial for him or her? ($2.718) \. PROBLEMS: do%)
The time it takes for a student to ﬁnish an aptitude test (in hours), say X, has a density function of the form
f(x) = { (a) Probability {l<X<1.5} =‘? Why does “1<X<l.5” have a probability? (6%)
(b) Var{X] 3? (4%) dx—DQ—x)ﬁl<x<2 0 elsewhere. GO ON TO THE NEXT PAGE. NTUEE Probability 8; Statistics 2006 Spring PROBLEM 9: (15%)
Random Number Generation
(a) Suppose X is a random variable with a cumulative distribution function F. Suppose that
i F is continuous. Define Y = F(X). Prove that Y has the uniform distribution on (0: 1).
That is, Show that Y satisﬁes P(Y S y) 2 y for all y€(0, l). (8%) (b) You have a computer code that generates random numbers uniformly distributed in (O, 1].
Let the corresponding random variable be X. Now you want to generate a random
variable Y of Gamma distribution with parameters (2, it), where 700. Describe how
you may apply (a) to generate Y by using the given uniform random number generator code and explain Why. (7%) PROBLEM 10: (15%) ‘ Some city draws electricity completely from a given power plant. The city is connected to this
power plant with two power lines. The times before failure for these two lines are independent.
They are exponentially distributed with parameters M and 12 (hour—1), respectively. Now,
both lines are fine at time 0. (a) One of these two lines fails ﬁrst at time t. Find the distribution of t. (5%) (b) Find the probability that the ﬁrst failed line is line 1. (5%) (c) Find the distribution of t given that it is known that line 1 fails first. (5%) PROBLElel: (10%)
A point (X, Y) is selected at random from a triangle whose three vertices are (0, O), (l, l), and (1, l) on thex —y plane.
(a) Find the joint probability density function of X and Y. (5%)
(b) Find the probability density function of X + Y. (5%) STOP
PLEASE RECHECK YOUR ANSWERS! THANK YOU! LL) ...
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This note was uploaded on 02/21/2012 for the course EE 101 taught by Professor 張捷力 during the Spring '07 term at National Taiwan University.
 Spring '07
 張捷力

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