Q1: PDF Suppose we make parts on a production line and measure the amount Y by...

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Q1: PDF Suppose we make parts on a production line and measure the amount Y by which the weight of the part differs from the target weight. From experience, we know that the distribution of Y is given by the probability density function where f(y) takes value 0 elsewhere. a. Find the constant c. f(y) = c(1 y), 1 y 1, b. Find the cumulative distribution function F(y). c. Find P(Y < 0.5). d. Find the expected value of Y, y. e. Find the standard deviation of Y, y. [3 marks] [5 marks] [2 marks] [2 marks] [3 marks] [Question total: 15] Q2: Expected reciprocal Let Y be uniformly distributed on the interval (1, 2). a. Find E(1/Y). b. Is E(1/Y) = 1/E(Y)? Justify your answer. [2 marks] [2 marks] [Question total: 4] Q3: Transformation of random variables a. Let Y be a normally distributed variable with mean and variance o. Let Z = 2Y. Using the method of transformation, what is the pdf of Z? b. Hence, describe the distribution of Z [5 marks] [2 marks] [Question total: 7] Q4: Transformation of random variables The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C 100+40Y+3Y2 relates the cost C of completing this operation to the square of the time to completion. a. Find the mean of C. b. Find the variance of C. [6 marks] [6 marks] [Question total: 12] Q5: Poisson process Buses arrive at a city according to a Poisson process with a rate of 4 per hour. a. What is the probability that no buses arrive between 11 a.m. and 1 p.m. on a randomly selected day. [2 marks] b. Given no buses have arrived in the two hours, what is the probability that one arrives in the next 1/2 an hour? [3 marks] c. What is the probability that the third bus of the day arrives after midday given they start arriving at 9 a.m. d. What is the probability that at most 2 buses arrive in the first hour? [2 marks] [2 marks] Q1: PDF Suppose we make parts on a production line and measure the amount Y by which the weight of the part differs from the target weight. From experience, we know that the distribution of Y is given by the probability density function where f(y) takes value 0 elsewhere. a. Find the constant c. f(y) = c(1 y), 1 y 1, b. Find the cumulative distribution function F(y). c. Find P(Y < 0.5). d. Find the expected value of Y, y. e. Find the standard deviation of Y, y. [3 marks] [5 marks] [2 marks] [2 marks] [3 marks] [Question total: 15] Q2: Expected reciprocal Let Y be uniformly distributed on the interval (1, 2). a. Find E(1/Y). b. Is E(1/Y) = 1/E(Y)? Justify your answer. [2 marks] [2 marks] [Question total: 4] Q3: Transformation of random variables a. Let Y be a normally distributed variable with mean and variance o. Let Z = 2Y. Using the method of transformation, what is the pdf of Z? b. Hence, describe the distribution of Z [5 marks] [2 marks] [Question total: 7] Q4: Transformation of random variables The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C 100+40Y+3Y2 relates the cost C of completing this operation to the square of the time to completion. a. Find the mean of C. b. Find the variance of C. [6 marks] [6 marks] [Question total: 12] Q5: Poisson process Buses arrive at a city according to a Poisson process with a rate of 4 per hour. a. What is the probability that no buses arrive between 11 a.m. and 1 p.m. on a randomly selected day. [2 marks] b. Given no buses have arrived in the two hours, what is the probability that one arrives in the next 1/2 an hour? [3 marks] c. What is the probability that the third bus of the day arrives after midday given they start arriving at 9 a.m. d. What is the probability that at most 2 buses arrive in the first hour? [2 marks] [2 marks]