homework4-winter11

homework4-winter11 - Homework 4. • • • • • 1....

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Unformatted text preview: Homework 4. • • • • • 1. ­ A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. Determine the probability mass function of X, the total dollar value of all sales. 2. ­ Four independent flips of a fair coin are made. Let X denote the number of heads obtained. Plot the probability mass function of the random variable X ­2. 3. ­ A gambling book recommends the following “winning strategy” for the game of roulette. It recommends that a gambler bet 1 on red. If red appears (which has probability 18/38), then the gambler should take her 1 profit and quit. If the gambler loses this bet (which has probability 20/38 of occurring), she should make additional 1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler’s winnings when she quits. (a) Find P(X > 0) (hint: write first the probability mass function of X). (b) Are you convinced that the strategy is indeed a “winning” strategy? Explain your answer. (c) Find E(X). Homework must be answered in the order shown here (else please make a note telling the reader where it is). Write your answers neatly Work must be shown for full credit No late homework accepted under any circumstances Homework must be stapled 4. ­ If E(X) = 1 and Var(X) = 5, find (a) E[(2+X)2] (b) Var(4+3X) 1 5. The manager of a stock room in a factory knows from his study of records that the daily demand (number of times used) for a certain tool has the following probability distribution: Quantity demanded Probability 0 0.1 1 0.5 2 0.4 (a) (In other words, 50 % of the daily records show that the tool was use one time). Let X denote the daily demand. (a) How much can would he expect the tool to be used tomorrow? (b) By how much could he be off? (c) Suppose that it costs the factory $10 each time the tool is used. What is the expected daily cost of using the tool? How much is this cost expected to vary from one day to another? 6.- It is known that diskettes produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the diskettes in packages of size 10 and offers a money-back guarantee that at most 1 of the 10 diskettes in the package will be defective. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them? 7. The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that there will be (a) at least 2 such accidents in the next month? (b) at most 1 accident in the next month? 8.. ­ The number of times that a person contracts a cold in a give year is a Poisson random variable with parameter λ=5. Suppose that a new wonder drug (based on large quantitites of vitamin C) has just been marketed that reduces the Poisson parameter to λ=3 for 75% of the population. For the other 25 percent of the population the drug has no appreciable effect on colds. If an individual tries the drug for a year and has 2 colds in that time, how likely is it that the drug is beneficial for him or her? 2 ...
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