ECEn 370 Introduction to Probability

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1 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 on both sides. Graphing or Scientic Calculator Allowed -Hour Suggested Time Limit IMPORTANT! WRITE YOUR NAME on every page of the exam. Answer questions -6 on the provided bubble sheet. Questions -6 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 ECEn 7 Midterm. Let A and B be two independent events in S. It is known that P (A B) =.9 and P (A B) =.5. Calculate P (A). A).6 B). C).5 D). E).5 F).4 G).45 H).5 I).64. Out of the students in a class, 5% are tall, % are athletes, and % fall into both categories. Determine the probability that a randomly selected student in this class is neither tall nor an athlete. A) % B) % C) % D) 4% E) 5% F) 6% G) 7% H) 8% I) 9%. A batch of forty items is inspected by testing four randomly selected items. If one of the four is defective, the batch is rejected. What is the probability that the batch is accepted if it contains ve defectives? A).8 B).5 C).8 D).49 E).49 F).5 G).57 H).647 I).8 4. Sixty 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).7 D). E).8 F). G).9 H).966 I).98

3 ECEn 7 Midterm 5. A power utility can supply electricity to a city from dierent power plants (not very good ones). Each power plant fails with probability.6. Suppose that two power plants are necessary to keep the city from a black-out. Find the probability that the city will experience a black-out. A).4 B).6 C). D).4 E).46 F).6 G).6 H).4 I) A number source generates digits,, and. The probability of a is., of a is.6, and of a is.. A ve-digit sequence is generated. What is the probability that two s will occur in the sequence? A). B).5 C). D).5 E). F).7 G).6 H).68 I) A random variable is called a Rayleigh random variable if its pdf is given by { x f X (x) = σ e x /(σ ) x > x < The radial distance [in meters (m)] of the landing point of a parachuting sky diver from the center of a target area is known to be a Rayleigh random variable with parameter σ. If the probability that the sky diver will land within a radius of m from the center of the target area is., what is σ? A) < σ < B) σ < C) σ < D) σ < 4 E) 4 σ < 5 F) 5 σ < 6 G) 6 σ < 7 H) 7 σ < 8 I) 8 σ.

4 ECEn 7 Midterm 8. 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 third attempt, what is the probability that the computer successfully sends the message on the second attempt? A).4 B).4 C).6 D).75 E).6 F).65 G).775 H).98 I) Suppose that this year the probability of at least one date is.9, the probability of at least one missing homework is.6, and the probability of at least one bombed test is.. What is the probability that this year you will have at least one date or at least one missing homework or at least one bombed test? A). B).8 C).4 D).648 E).7 F).89 G).9 H).968 I).98. A random variable X has a probability density function given by f X (x) = { (x ) 4, if x 5,, otherwise. Find the cumulative distribution function, F X (x), and compute the value of the following expression: A) B) /6 C) /4 D) 7/6 E) F) /6 G) 4 H) 54 I) 97/6 F X () + F X (4) + F X (9) 4

5 ECEn 7 Midterm. You have the following joint PMF of random variables X and Y : y 4 / / / / 4 5 x Find E[Z] where Z = XY A).58 B).96 C). D).9 E) 6.8 F) 7.4 G) 7.5 H).9 I) 4.9. 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 (max(x, Y, Z) = 4). A).5 B).6 C).7 D).7 E).64 F).9 G).5 H).6 I).4. Suppose that X, Y, and Z are independent random variables. E[X] = 4, E[Y ] =, and E[Z] =. var(x) =, var(y ) =, and var(z) =. Compute the following expressions. E[XY Z Y ] + var(x Y ) + E[XY ] var( Z) A) 8 B) C) D) 4 E) 8 F) 4 G) 5 H) 5 I) 6 5

6 ECEn 7 Midterm 4. The new fad of Greenie Babies started. There are ve plush animals. Stores of McTacoKing will give you one of the ve with each meal purchase, but you don't know which one you will get. What is the expected number, n, of McTacoKing meal purchases you need to make to collect them all? A) < n 5 B) 5 < n 7 C) 7 < n 9 D) 9 < n E) < n F) < n 5 G) 5 < n 7 H) 7 < n 9 I) 9 < n 5. 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] + E[Y ] x A) 7/6 B) 4/ C) / D) 5/ E) /6 F) G) /6 H) 7/ I) 5/ 6. A binary signal S is transmitted, and we are given that P (S = ) =.6 and P (S = ) =.4. 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.7 of Y? A).4 B).67 C).5 D).7 E).677 F).7 G).7 H).8 I).859 6

7 ECEn 7 Midterm 7. The time until a small meteorite rst lands anywhere in the Sahara desert is modeled as an exponential random variable with a mean of days. The time is currently midnight. What is the probability that a meteorite rst lands some time between 6 a.m. and 6 p.m. of the rst day? A).5 B).5 C).48 D).7 E).8 F).8 G).9 H).97 I).9 8. 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 Compute the following: c + 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 x 9. From the gure above in problem 8, 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 (, ) + F X,Y (.5, 4.5) + F X,Y (4.5, 6) A) 9/6 B) /6 C) /6 D) /6 E) /6 F) 4/6 G) 5/6 H) 6/6 I) 7/6 7

8 ECEn 7 Midterm. Suppose that the random variable X has the piecewise constant PDF /, if x, f X (x) = /, if < x, otherwise Evaluate the following: E[X] + var(x) A) 5/ B) /6 C) D) /6 E) 7/ F) 5/ G) 8/ H) 7/6 I). 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 share 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, y x Find the mean of the trader's prots (or losses if negative). A) - B) -/ C) D) / E) F) / G) H) 5/ I) 8

9 ECEn 7 Midterm. Let X be a random variable with PMF Find a + E[X] + var(x) A) 7 B) 9 C) 9 D) 4 E) 5 F) 6 G) 76 H) 77 I) 78 p X (x) = { x /a, if x =,,,,,,,, otherwise 4. Let X be a random variable that takes values from to 9 with equal probability /. Find the PMF of the random variable Y = X mod(). Calculate p Y () + p Y () A) / B) / C) / D) 4/ E) 5/ F) 6/ G) 7/ H) 8/ I) 9/ 5. 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).896 B).898 C).9 D).9 E).94 F).96 G).98 H).9 I).9 9

10 ECEn 7 Midterm 6. The scores on an exam are normally distributed with a mean of 8 and a standard deviation of. If you know your score, x, is above 75% of your peers (but 5% of your peers are above you), then choose the correct range where your score falls. NOTE: Values of the standard normal CDF are Φ() =.5, Φ(.) =.579, Φ(.4) =.6554, Φ(.6) =.757, Φ(.8) =.788, Φ(.) =.84 A) < x < 65 B) 65 < x < 7 C) 7 < x < 75 D) 75 < x < 8 E) 8 < x < 85 F) 85 < x < 9 G) 9 < x < 95 H) 95 < x < I) x >

ECEn 370 Introduction to Probability

ECEn 370 Introduction to Probability 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.