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### 20115ee131A_1_ps3_2011fall

Course: EE 132A, Fall 2011
School: UCLA
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Word Count: 460

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131A Probability Instructor: EE Lara Dolecek Problem Set 3 Wednesday, October 12, 2011 Due: Wednesday, October 19, 2011 Reading: Chapters 3 and 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia 100 points total 1. Drawing dollars from an urn, without replacement. Problem 3.7, page 131 of ALG. An urn contains 9 \$1 bills and one \$50 bill. Let the random variable X be the total amount that...

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131A Probability Instructor: EE Lara Dolecek Problem Set 3 Wednesday, October 12, 2011 Due: Wednesday, October 19, 2011 Reading: Chapters 3 and 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia 100 points total 1. Drawing dollars from an urn, without replacement. Problem 3.7, page 131 of ALG. An urn contains 9 \$1 bills and one \$50 bill. Let the random variable X be the total amount that results when two bills are drawn from the urn without replacement. (a). Describe the underlying space S of this random experiment and specify the probabilities of it elementary events. (b). Show the mapping from S to SX , the range of X . (c). Find the probabilities for various values of X . 2. Drawing dollars from an urn, with replacement. Problem 3.8, page 131 of ALG. An urn contains 9 \$1 bills and one \$50 bill. Let the random variable X be the total amount that results when two bills are drawn from the urn with replacement. (a). Describe the underlying space S of this random experiment and specify the probabilities of it elementary events. (b). Show the mapping from S to SX , the range of X . (c). Find the probabilities for various values of X . 3. Mean and variance. Find the mean and variance of X in problem 1 and in problem 2. 4. Probabilities. What is the probability of drawing more than 2 dollars in problem 1 and in problem 2? 5. St. Petersburg Paradox. Problem 3.34, page 133 of ALG. Consider St. the Petersburg paradox in Example 3.16. Suppose that the casino has a total of M = 2m dollars, and so it can only aord a nite number of coin tosses. (a). How many tosses can the casino aord? (b). Find the expected payo to the player. (c). How much should a player be willing to pay to play this game? 6. Transformations of an RV. Let X be a uniform discrete RV taking values in the set {0, 1, 2, 3, 4}. Let Y = X + 100 and let Z = 10X . Compute mean and variances of X, Y and Z . 1 7. Higher order moments. Compute third and fourth moment of X in problem 6. 8. Packet transmission. In data transmission of 10 bytes, a byte error occurs independently with probability 1/10. What is the probability that in a 10-byte transmission, less than 3 byte errors occur ? 9. Poisson approximation to binomial. Problem 3.63, page 136 of ALG. Compare the Poisson approximation and the binomial probabilities for k = 0, 1, 2, 3 and n = 10, p = 0.1; n = 20 and p = 0.05; and n = 100 and p = 0.01. 10. Cdf warm up. An urn contains 8 \$1 bills and two \$5 bills. Let X be the total amount that results when two bills are drawn from the urn without replacement, and let Y be the total amount that results when two bills are drawn from the urn with replacement. (a). Plot and compare the cdfs of the random variables. (b). Use the cdfs to compare p{X = \$2} and p{Y = \$2}; p{X < \$7} and p{Y < \$7}; p{X \$6} and p{Y \$6}. 2
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UCLA - EE - 132A
EE 131AProbabilityInstructor: Lara DolecekProblem Set 3Wednesday, October 12, 2011Due: Wednesday, October 19, 2011Reading: Chapters 3 and 4 of Probability, Statistics, and Random Processes by A.Leon-Garcia100 points total1. Drawing dollars from a
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UCLA - EE - 132A
EE 131AProbabilityInstructor: Lara DolecekProblem Set 4Wednesday, October 19, 2011Due: Wednesday, November 2, 2011Reading: Chapter 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia100 points total1. CDF, PDF and median. A PDF of
UCLA - EE - 132A
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UCLA - EE - 132A
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UCLA - EE - 132A
EE 131AProbabilityInstructor: Lara DolecekProblem Set 6Wednesday, November 9, 2011Due: Wednesday, November 16, 2011Reading: Chapter 5 of Probability, Statistics, and Random Processes by A. Leon-Garcia100 points total1. Coin tossing and two RVs. Pr
UCLA - EE - 132A
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