06 STAT HWA q continuous rv - Dr. V.R. Bencivenga Economics...

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Dr. V.R. Bencivenga Economics 329 PRACTICE HOMEWORK #6B: CONTINUOUS RANDOM VARIABLES 1. Cities A and C are 60 miles apart by highway. City B is on the highway between A to C, 20 miles from A. If a car has a breakdown on the highway between A and C, the location of the breakdown can be described by a uniformly distributed random variable X. An automobile service (AAA) has contracts with garages in cities A, B, and C, and it will tow AAA members to the nearest garage for repairs, in the event of a breakdown. a. What is the probability a car that breaks down will need to be towed more than 5 miles? b. What is the probability a car will be need to be towed more than 5 miles, given that it is more than 10 miles from City A? c. Suppose you own the garages in Cities A, B, and C with which AAA has contracts. What proportions of revenues from towing AAA members are earned by each of your garages? d. Your variable cost is $1 per mile driven to a car that has broken down (round-trip mileage). What is your expected cost per breakdown ? e. Suppose you could “start over” and locate your three garages anywhere along this 60 mile stretch of highway. Where would you locate them? Explain. 2. You own a small business. Your monthly profits depend on revenues, which range from $5 thousand to $10 thousand, and costs, which range from $3 thousand to $7 thousand. Assume that revenues and costs are jointly uniformly distributed and independent. a. What are your expected monthly profits? b. What is the probability that your profits will be negative in any given month? c. What is the probability that your monthly profits will be negative given that your revenues are less than $7 thousand? d. Suppose your monthly costs increase; specifically, now costs range from $3 thousand to $8 thousand. Assuming that your revenues and cost continue to be jointly uniformly distributed, and that your minimum monthly revenue continues to be $5 thousand, by how much would the upper end of the range of revenues need to increase in order for the probability of negative profits not to increase? 3. For each part below, sketch the pdf’s of the normal random variables X and Y on the same graph. For each pdf, label the values of the random variable corresponding to the mean and one standard deviation. Also, for each part, state Y as a linear transformation of X. a. X has a mean of 2 and a variance of one; Y has a mean of zero and a variance of one. b. X has a mean of zero and a variance of .25; Y has a mean of zero and a variance of one. c.
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06 STAT HWA q continuous rv - Dr. V.R. Bencivenga Economics...

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