realquiz2sol - PROBABILITY AND DECISION MAKING 45-730 QUIZ...

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PROBABILITY AND DECISION MAKING 45-730 QUIZ 2 (OFFLINE) Problem 1 ( Expectation ) . Consider the following random variables: The random variable X takes the values { 0 , 1 , 2 } with equal probabilities. The random variable Y takes the value 9 with probability 1 3 and 12 with probability 2 3 . The random variable Z takes the values { 2 , 4 , 6 , 8 } with equal probabilities. (1.1) What is the expected value of X ? Solution. E [ X ] = 1 3 · 0 + 1 3 · 1 + 1 3 · 2 = 1. / (1.2) What is the expected value of Y ? Solution. E [ Y ] = 1 3 · 9 + 2 3 · 12 = 11. / (1.3) What is the expected value of Z ? Solution. E [ Z ] = 1 4 · 2 + 1 4 · 4 + 1 4 · 6 + 1 4 · 8 = 5. / (1.4) What is the expected value of the random variable W = 2 X - Y + 4 Z ? Solution. By the linearity of expectation we have that E [ W ] = E [2 X - Y + 4 Z ] Therefore E [ W ] = 2 · E [ X ] - E [ Y ] + 4 · E [ Z ] = 2 - 11 + 20 = 11. / Problem 2 ( Variance ) . Consider the following four random variables. Which of these has the highest variance? A: Binomial random variable with parameters p = 1 2 ,n = 4. B: Poisson random variable with mean = 1. C: Discrete random variable which taking one of the values 0 , 1 , 2 , 3 , 4 with equal probabilities. D: Discrete random variable which takes value 0 with probability 1 2 and value 4 with probability 1 2 . (2.1) What is the variance of A ? Solution. V ar ( A ) = 4 · 1 2 · 1 2 . / (2.2) What is the variance of B ? Solution. V ar ( B ) = 1. / (2.3) What is the variance of C ? Solution. V ar ( C ) = 1 5 · [2 2 + 1 2 + 0 2 + 1 2 + 2 2 ] = 2. / (2.4) What is the variance of D ? Solution. V ar ( D ) = 1 2 · [2 2 + 2 2 ] = 4. / Problem 3 ( ) . A vending machine sells snacks at $0 . 90 each. What is the expected value and variance of daily revenue Y from the machine, if X , the number of snacks sold per day, has E ( X ) = 230 and V ar ( X ) = 40 ? (3.1) Can we compute the expected value of Y ? If so, what is its value? Solution. We have that the random variable Y is given by Y = 0 . 90 · X . So E ( Y ) = 0 . 90 · E ( X ) = 0 . 90 · 230 = 207. / (3.2) Can we compute the variance of Y ? If so, what is its value? 1
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2 QUIZ 2 (OFFLINE) Solution. V ar ( Y ) = 0 . 90 · 0 . 90 · V ar ( X ) = 0 . 90 · 0 . 90 · 40 = 32 . 40. / (3.3) Let T be a random variable with variance V ar ( T ) = 20 and the covariance of X and T is Cov ( X,T ) = 30. Can we compute the variance of the random variable Z = 3 X - T ? If so, what is its value? Solution.
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This note was uploaded on 10/05/2010 for the course BUS 45730 taught by Professor Ravi during the Spring '08 term at Carnegie Mellon.

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realquiz2sol - PROBABILITY AND DECISION MAKING 45-730 QUIZ...

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