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STAT4203_Homework3 - with mean 50 minutes and standard...

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Fall 2007 STAT 4203 Homework 3 Due on October 1, 2007 before class 1. The PH of water samples from a specific lake is a random variable Y with the following density function: - = . , 0 7 5 , ) 7 ( ) ( 2 elsewhere y y c y f a. Find the value of c that makes f(y) a probability density function. b. Find the cumulative probability function of y, i.e. F(y). c. Find E[Y] and Var(Y). d. Would you expect to see a PH measurement below 5.5 very often? Why? 2. Suppose a deficiency is randomly located on a cable line. If one randomly selects a 10-foot long cable, what’s the probability that the deficiency occurs: a. At exactly 2 feet of the beginning of the cable line? b. Within 2 feet of the beginning of the cable line? 3. The length of time required to complete a test is found to be normally distributed
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Unformatted text preview: with mean 50 minutes and standard deviation 10 minutes. When should the test be terminated if we wish to allow sufficient time for 95% of the students to complete the test? 4. Suppose the length of time (Y) necessary to fix a copier machine follows an exponential distribution with mean 2 hours. The formula C=10+5Y + 2Y 2 relates the repair cost C and Y. Find the mean and variance of C. 5. If ) 1 , 1 ( ~ Beta Y , show Y ~ Uniform(0,1). That is, the uniform distribution over interval (0,1) is a special case of a beta distribution. 6. If a random variable Y has the following density: +∞ < < ∞-=-y ke y f y , ) ( 2 / 2 a. Find k such that f(y) is a valid pdf. b. Find the moment-generating function of Y. c. Find E[Y] and Var(Y) by moments....
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