ECEn 370 Introduction to Probability

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1 RED- You can write on this exam. ECEn 7 Introduction to Probability Section Midterm Winter, Instructor Professor Brian Mazzeo Closed Book - You can bring one 8.5 X sheet of handwritten notes on both sides. Graphing or Scientic Calculator Allowed -Hour Suggested Time Limit IMPORTANT! WRITE YOUR NAME on every page of the exam. Answer questions -7 on the provided bubble sheet. Questions -7 are worth point each. Do not discuss the exam with other students. NOTE: Use all of the digits on your calculator, or fractions, before at the end rounding to the number of signicant digits used in the problem.

2 . A batch of fty items is inspected by testing three randomly selected items. If one of the three is defective, the batch is rejected. What is the probability that the batch is accepted if it contains seven defectives? A). B).4 C).8 D).49 E).49 F).6 G).66 H).676 I).8. Ninety students, including Joe and Jane, are to be split into three classes of equal size, and this is to be done at random. What is the probability that Joe and Jane end up in the same class? A).6 B).7 C). D).6 E).8 F). G).9 H).966 I).98. If at rst you don't succeed, try, try, try again. A computer will successfully send a message across a network with probability.6. The computer will retry sending the message until it is successfully sent. Given that we know that the computer will successfully transmit the message on or before the fourth attempt, what is the probability that the computer successfully sends the message on the rst attempt? A).4 B).4 C).6 D).75 E).66 F).65 G).775 H).98 I).946

3 4. A random variable X has a probability density function given by ( fx (x) = (x ), 4 if x 4, otherwise., Find the cumulative distribution function, FX (x), and compute the value of the following expression: FX () + FX () + FX (9) A) B) /6 C) /4 D) 7/6 E) F) /6 G) 4 H) 54 I) 97/6 5. You have the following joint PMF of random variables X and Y : y 4 / / / 4 5 / Find E[Z] where Z = XY A).58 B).96 C). D).9 E) 6.8 F) 7.4 G).76 H) 4.8 I) 4.9 x

4 6. You have three ten-sided dice numbered from to. Each face has an equal probabilitiy of appearing during a roll (uniformly distributed). The outcomes, representing each die, are the random variables X, Y, and Z. Find P (min(x, Y, Z) = 8). A).5 B).9 C).7 D).7 E).64 F).9 G).5 H).6 I).4 7. You have the following joint PDF with a constant density, f X,Y (x, y) = c, in the shaded region and is zero outside that region: y Find E[X ] x A) / B) 4/ C) / D) 5/ E) /6 F) G) /6 H) 7/ I) 5/ 4

5 8. A binary signal S is transmitted, and we are given that P (S = ) =.7 and P (S = ) =.. The received signal is Y = N + S, where N is normal noise, with zero mean and unit variance, independent of S. What is the probability that S =, as a function of the observed value of -. for Y? A).4 B).6 C).44 D).656 E).677 F).7 G).74 H).8 I) We are told that the joint PDF of the random variables X and Y is a constant c on the set S shown in the gure below and is zero outside. We wish to determine the value of c and the marginal PDFs of X and Y. y 4 S x Compute the following: c + f Y (.5) + f Y (.5) + f Y (.5) + f X (.5) + f X (.5) + f X (.5) A) B) 9/4 C) 5/ D) /4 E) F) /4 G) 7/ H) 5/4 I) 4 5

6 . From the gure above in problem 9, compute the following where F X,Y is the CDF of the joint PDF. F X,Y (, ) + F X,Y (.5,.5) + F X,Y (.5,.5) + F X,Y (,.5) + F X,Y (.5, 4.5) + F X,Y (4.5, 6) A) /6 B) /6 C) /6 D) 4/6 E) 5/6 F) 6/6 G) 7/6 H) 8/6 I) 9/6. Each morning, Hungry Hungry Horace eats some sausages. On any given morning, the number of sausages he eats is equally likely to be,, 8, or 9, independent of what he has done in the past. Let X be the number of sausages that Harry eats in days. Compute q = E[X] + var(x). A) < q B) < q C) < q 5 D) 5 < q 7 E) 7 < q 9 F) 9 < q G) < q H) < q 5 I) 5 < q 7. A stock market trader buys shares of stock A and shares of stock B. Let X and Y be the price changes of A and B, respectively, over a certain time period, and assume that the joint PMF of X and Y is uniform over the set of integers x and y satisfying x, x y Find the mean of the trader's prots (or losses if negative). A) - B) -/ C) D) / E) F) / G) H) 5/ I) 6

7 . Let X be a random variable that takes values from to 9 with equal probability /. Find the PMF of the random variable Y = min ( X, X 4, X 5 ). Calculate p Y () + p Y () A) / B) / C) / D) 4/ E) 5/ F) 6/ G) 7/ H) 8/ I) 9/ 4. Al thows darts at a circular target of radius m and is equally likely to hit any point on the target. Let X be the distance of Al's hit from the center. Find E[X] + var(x). A).8 m B).8 m C).85 m D).87 m E).89 m F).8 m G).8 m H).85 m I).87 m 5. Let A, B, and C be three events in Ω. If P (A) =, P (B) = 4, P (C) =, P (A B) = 8, P (A C) = 6, and P (B C) =, nd P (A B C). A) 6/4 B) 7/4 C) 8/4 D) 9/4 E) /4 F) /4 G) /4 H) /4 I) 7

8 6. Suppose you are in a Family Home Evening Group with ten men and ten women. On any given Monday night, the probability of a boy showing up is 8/ and the probability of a girl showing up is 9/, independently of any one else. What is the probability, P, that less than two people show up? A) < P < B) < P < C) 4 < P < D) 5 < P < 4 E) 6 < P < 5 F) 7 < P < 6 G) 8 < P < 7 H) 9 < P < 8 I) < P < 9 7. Suppose I have a uniform random variable, X, with the following PMF: { /5, if x =, 4, 5, 6, 7 p X (x) =, otherwise. Suppose I have a random variable, Z = (X 4). What is E[Z]? A) - B) -/ C) D) / E) F) / G) H) 5/ I) 8. Messages transmitted by a computer in Provo through a data network are destined for Salt Lake City with probability.5, Las Vegas, with probability., and for Denver with probability.. The transit time X of a message is random. Its mean is.5 seconds if it is destined for Salt Lake City,. seconds if it is destined for Las Vegas, and. seconds if it is destined for Denver. Calculate E [X]. A).45 B).45 C).45 D).45 E).5 F).5 G).45 H) 4.5 I) 45 8

9 The following conditional PMF is used for problems 9 and. You have the following conditional PMF of random variable Y conditioned on random variable X: y 4 /4 / / /4 4 5 x and marginal PMF of X given by: /6, if x =,,, 4 p X (x) = /, if x = 5, otherwise. The above conditional PMF is used for problems 9 and. 9. For the conditions above, compute E[Z], where Z = XY. A) B) C) D) 4 E) 5 F) 6 G) 7 H) 8 I) 9. Find the probability P (min(x, Y ) = ). A) /4 B) /8 C) /6 D) / E) / F) 7/ G) / H) /4 I) 9

10 . A proportion, p, of BYU students have served missions. I question N BYU students and I sum up the number that have served missions. Since I can't talk to all of the students, to estimate the proportion I then take my sum and divide by N to form a sample average to estimate p. Mark on your answer sheet all of the statements that are true (you can have multiple bubbles lled in on this one, if necessary). A) The expected value of the sample average decreases as N increases. B) The expected value of the sample average is the same as N increases. C) The expected value of the sample average increases as N increases. D) The variance of the sample average decreases as N increases. E) The variance of the sample average stays the same as N increases. F) The variance of the sample average increases as N increases. G) If N is, then the expected value of the sample average is or, depending on p. H) If N is, then the expected value of the sample average is p. I) If I ask a dierent set of N BYU students, I will always get the same sample average. J) None of the above are true.. Let X be the roll of a fair six-sided die and let A be the event that the roll is an even number. Find p X A (4). A) / B) / C) / D) 4/ E) 5/ F) 6/ G) 7/ H) 8/ I) 9/. The PMF for X is given by p X (x) = Let Y = X. What is p Y ( ) + p Y () + p Y () + p Y ()? A) /9 B) /9 C) /9 D) 4/9 E) 5/9 F) 6/9 G) 7/9 H) 8/9 I) { /9, if x is an integer in the range [ 4, 4],, otherwise

11 4. A parking lot contains cars, of which happen to be lemons. We select 4 of these cars at random and take them for a test drive. Find the probability that of the cars tested turn out to be lemons. A) 5. 5 B).4 C).4 D) 4. E) 4.6 F) 4.86 G) 5. H) 8. I) 4. For problems 5 and 6, A source transmits a message (a string of symbols) through a noisy communication channel. Each sysmbol is or with probability.4 and.6, respectively, and is received incorrectly with probability. and.7, respectively. Errors in dierent symbol transmissions are independent. 5. What is the probability that the string of symbols is received correctly? A).8 B).89 C).56 D).84 E).44 F).864 G).9 H).4 I) In an eort to improve reliability, each symbol is transmitted three times and the received string is decoded by majority rule. In orther words, a (or ) is trasmitted as (or, respectively), and it is decoded at the receiver as a (or ) if and only if the received three-symbol string contains at least two s (or s, respectively). What is the probability that a is correctly decoded? A).44 B).44 C).445 D).447 E).449 F).45 G).45 H).455 I).457

12 7. This is a dicult problem for the last one. The joint pdf of random variables X and Y is given by { y f X,Y (x, y) = e x/y e y x >, y >, otherwise Find P (X > Y = ). A).45 B).47 C).49 D).5 E).5 F).55 G).57 H).59 I).6

ECEn 370 Introduction to Probability

ECEn 370 Introduction to Probability ECEn 7 Midterm RED- You can write on this exam. ECEn 7 Introduction to Probability Section Midterm Winter, Instructor Professor Brian Mazzeo Closed Book - You can bring one 8.5 X sheet of handwritten notes