ECEn 370 Midterm Name:

RED- You can write on this exam.

ECEn 370

Introduction to ProbabilitySection 001

Midterm

Winter, 2010

Instructor

Professor Brian Mazzeo

Closed Book

Testing Center Calculator

3-Hour Time Limit

IMPORTANT!

• WRITE YOUR NAME on every page of the exam.

• Answer questions 1-20 (Part I) on the provided bubble sheet.

• Questions 1-20 are worth 1 point each.

• Answer question 21 (Part II) directly on the exam.

• Question 21 has its points speci�ed individually.

• Do not discuss the exam with other students.

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ECEn 370 Midterm Name:

Part I

1. A test for the probability-phobia-disease is assumed to be correct 95% of the time. We also assume that theprobability of any random person in our class having this disease is 0.01. Given that you just tested positive forthis disease, what is the probability that you do not have the disease?

A) 0.9900B) 0.9000C) 0.8390D) 0.7250E) 0.0500F) 0.0100G) 0.0099H) 0.0098I) 0.0001J) None of the Above

2. I have the sets A, B, C which are events formed from a sample space, Ω. Which of the following statements arealways true?

I) A ∪ (A ∩B) = A ∩ (A ∪B)II) A ∩Ac = ∅III) (A ∩B)c = Ac ∪BIV) A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)V) Ac = ∅.

A) I only.B) I and II only.C) I and III only.D) I, II, and IV only.E) II and III only.F) I, II, III, and V only.G) IV only.H) II and IV only.I) I, II, III, IV, and V.J) None of the Above.

3. It is Friday and after ECEn 370 you want to �nd a date for tonight and there are two people, A and B, youwould like to ask. The probability that A will accept an invitation is 310 . The probability that B will accept aninvitation is 23 . You randomly select one of the two and extend an invitation. If you are rejected, then you go tothe other person and extend an invitation. We make the following assumptions: (1) you can only end up with onedate tonight and (2) once you are rejected, you can't ask the same person again tonight. What is the probabilitythat you will have a date with person B tonight?

A) 1/5B) 2/3C) 3/10D) 17/30E) 23/30F) 17/23G) 7/10H) 8/9I) 1/3J) None of the Above.

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ECEn 370 Midterm Name:

4. Three students invite six people to an event tonight. The probability that any one of the six people will showup is 12 , independent of the others. What is the probability that more than three people show up?

A) 21/32B) 1/8C) 1/4D) 1/2E) 9/32F) 3/16G) 3/32H) 1/32I) 11/32J) None of the Above.

5. You have the following joint PMF of random variables X and Y :

1 2 3 4

1

2

3 1/10

y

x

1/10

1/10 1/10

1/10

1/10

1/10

1/102/10

0

0

0

Evaluate the following expression:E[X] + E[Y ] + var(X)

A) 5.4500B) 4.2000C) 3.6500D) 3.2500E) 2.7000F) 2.5000G) 1.7000H) 1.2000I) 0.7500J) None of the Above.

6. Suppose that X, Y , and Z are independent random variables. E[X] = 3, E[Y ] = 2, and E[Z] = 1. var(X) = 3,var(Y ) = 2, and var(Z) = 1. Evaluate the following expression:

E[2X +X2] + var(Y −X)

A) 5B) 7C) 10D) 11E) 16F) 17G) 23H) 25I) 28J) None of the Above.

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ECEn 370 Midterm Name:

7. Laman produces 1,000 widgets, 100 of which are defective. Lemuel produces 2,000 widgets, 150 of which aredefective. The widgets from both are combined and a widget is selected at random and found to be defective. Whatis the probability it came from Laman?

A) 0.0333B) 0.0500C) 0.0833D) 0.2000E) 0.2500F) 0.2333G) 0.3000H) 0.4000I) 0.4333J) None of the Above.

8. Of a group of 100 students, 20 students are women. Two students are selected at random, without replacement,from the group. What is the probability that the second student selected is a woman, given that the �rst studentselected was a woman?

A) 0.100B) 0.192C) 0.200D) 0.215E) 0.250F) 0.300G) 0.333H) 0.350I) 0.400J) None of the Above.

9. Let A and B be two independent events in S. It is known that P (A ∩B) = 0.16 and P (A ∪B) = 0.64.Calculate P (A).

A) 0.160B) 0.200C) 0.250D) 0.300E) 0.350F) 0.400G) 0.450H) 0.500I) 0.640J) None of the Above

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ECEn 370 Midterm Name:

10. A binary source generates digits 1 and 0. The probability of a 1 is 0.6 and the probability of a 0 is 0.4. A�ve-digit sequence is generated. What is the probability that two 1s and three 0s will occur in the sequence?

A) 0.100B) 0.150C) 0.230D) 0.250E) 0.300F) 0.317G) 0.600H) 0.683I) 0.700J) None of the Above.

11. The number of packets arriving at a server during any 1-second period is known to be Poisson distributed withλ = 2 arrivals per period. Find the probability that more than three packets will arrive during any 1-second period.

A) 0.933B) 0.124C) 0.135D) 0.140E) 0.143F) 0.148G) 0.149H) 0.150I) 0.153J) None of the Above.

12. A production line manufactures 1000-lb. weights that have 10 percent tolerance. Let X denote the weight of amanufactured weight. Assuming that X is a normal random variable with mean 1000 -b and variance 2500-lb, �ndthe probability that a weight picked at random will be rejected. Φ(y) is the cumulative distribution function of astandard normal random variable.

A) Φ(0.04)B) Φ(0.08)C) 1 - Φ(0.04)D) 1 - Φ(0.08)E) 2(1-Φ(0.04))F) 2(1 - Φ(0.08))G) 1 - Φ(2)H) 2(1 - Φ(2))I) 2(1 - Φ(4))J) None of the Above.

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ECEn 370 Midterm Name:

13. A random variable is called a Rayleigh random variable if its pdf is given by

fX(x) =

{xσ2 e

−x2/(2σ2) x > 00 x < 0

The radial distance [in meters (m)] of the landing point of a parachuting sky diver from the center of a target areais known to be a Rayleigh random variable with parameter σ2 = 100. Find the probability that the sky diver willland within a radius of 10 m from the center of the target area.

A) 0.393B) 0.396C) 0.398D) 0.400E) 0.402F) 0.405G) 0.408H) 0.410I) 0.413J) None of the Above.

14. If at �rst you don't succeed, try, try, try again. A computer will successfully send a message across a networkwith probability 0.7. The computer will retry sending the message until it is successfully sent. Given that we knowthat the computer will successfully transmit the message on or before the fourth attempt, what is the probabilitythat the computer successfully sends the message on the �rst attempt?

A) 0.2100B) 0.6221C) 0.6233D) 0.6943E) 0.7000F) 0.7057G) 0.7222H) 0.7234I) 0.8215J) None of the Above

15. You take the bus, walk, or ride your bike to school. In the morning you toss a fair, six-sided die.If you get a 1 or 2, or 3 you take the bus.If you get a 4, or 5 you walk.If you get a 6 you ride your bike.On average, it takes you 10 minutes if you take the bus, 15 minutes if you walk, and 6 minutes if you ride yourbike. What is the expected amount of time it takes you to get to campus?

A) 6 minutesB) 7 minutesC) 8 minutesD) 9 minutesE) 10 minutesF) 11 minutesG) 12 minutesH) 13 minutesI) 14 minutesJ) None of the Above.

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ECEn 370 Midterm Name:

16. A random variable X has a probability density function given by

fX(x) =

{2(x−5)

9 , if 5 ≤ x ≤ 8,0, otherwise.

Find the cumulative distribution function, FX(x), and compute the value of the following expression:

FX(1) + FX(7) + FX(9)

A) 0B) 1C) 13/9D) 5/3E) 16/9F) 17/9G) 19/9H) 8/3I) 3J) None of the Above

17. A joint probability density function is given by the following:

fX,Y (x, y) =

c, if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1,17 , if 2 ≤ x ≤ 3 and 2 ≤ y ≤ 7,0, otherwise.

What is the value of c that will make this a legitimate PDF?

A) 1/7B) 2/7C) 3/7D) 4/7E) 5/7F) 6/7G) 1/14H) 3/14I) 5/14J) None of the Above

18. You ask a person on a date over and over, and each time there is a probability 0.01 that he or she accepts,independent of previous attempts. While there are no guarantees that he or she will accept, how many tries is itexpected that it will take until he or she accepts?

A) 10B) 50C) 100D) 500E) 1000F) 5000G) 10000H) 0I) ∞J) None of the Above

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ECEn 370 Midterm Name:

19. A source transmits a string of symobls consisting of 0 and 1s through a noisy communication channel. Symbolsare received correctly except for the following errors:

Given that a 0 is transmitted, a 1 is received with probability 0.02.Given that a 1 is transmitted, a 0 is received with probability 0.01.

Errors in di�erent symbol transmissions are independent. If the string of symbols 00110 is transmitted, what is theprobability that it is received correctly?

A) 0.918B) 0.920C) 0.922D) 0.924E) 0.932F) 0.934G) 0.941H) 0.943I) 0.951J) None of the Above

20. Suppose that this year the probability of at least one date is 0.9, the probability of at least one missing home-work is 0.5, and the probability of at least one bombed test is 0.8. What is the probability that this year you willhave at least one date or at least one missing homework or at least one bombed test?

A) 0.04B) 0.09C) 0.36D) 0.50E) 0.72F) 0.80G) 0.90H) 0.95I) 0.96J) None of the Above

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ECEn 370 Midterm Name:

Part II

21. (5 points) Secret Diplomacy. You are hired by the State Department to analyze the probability of the followingtransactions. The CIA continually sends messages to the Ambassador in Tehran. To do this, the CIA makes twocopies of the message. They send one to Dick Tracy with probability of success 0.7 and the other to James Bondwith probability of success 0.8. Dick Tracy gets the message to the Ambassador with probability of success of0.8 and James Bond with probability of success of 0.9. The probability of success of each leg of the journey isindependent of the other legs.

a) What is the probability that a single message makes it to the Ambassador?

b) James Bond is being followed. If he has a message, there is a 0.3 probability that his message will be copiedwithout his knowledge and delivered to Mr. X (i.e. he may still deliver his message to the Ambassador). If youknow that Mr. X received a particular message from the CIA, what is the probability that the Ambassador receivedthe same message?

c) The CIA decides to send an important message repeatedly to make sure that the Ambassador gets it. What isthe probability that it takes three or less attempts to get the message to the Ambassador?

d) The CIA breaks up information to make sure if it falls into the hands of Mr. X, then he may not be able todecipher the total information. Suppose the CIA sends ten messages. Let Y be the number of these ten messagesthat fall into the hands of Mr. X. What is the mean of Y ?

e) What is the standard deviation of Y as de�ned in part d?

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