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Chapter 5: Q. 5.25 (page 216)
Let .
Show that if X is a Weibull random variable with parameters 谓, 伪, and 尾, then Y is an exponential random variable with parameter 位 = 1 and vice versa.
The above statement is proved.
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To be a winner in a certain game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on . If , then you are successful in round ; if , then you are successful in round ; and if , then you are successful in round .
(a) Find the probability that you are successful in round .
(b) Find the conditional probability that you are successful in round given that you were successful in round .
(c) Find the conditional probability that you are successful in round given that you were successful in rounds
(d) Find the probability that you are a winner
Let be a uniform random variable, and let be constants.
(a) Show that if, then is uniformly distributed on , and if , then is uniformly distributed on .
(b) Show that is uniformly distributed on .
(c) What function of is uniformly distributed on
(d) Show that is a uniform random variable.
(e) Show that is a uniform random variable.
Trains headed for destination A arrive at the train station at -minute intervals starting at 7 a.m., whereas trains headed for destination B arrive at -minute intervals starting at 7:05 a.m.
(a) If a certain passenger arrives at the station at a time uniformly distributed between and a.m. and then gets on the first train that arrives, what proportion of time does he or she go to destination A?
(b)What if the passenger arrives at a time uniformly distributed
between and a.m.?
(a) A fire station is to be located along a road of length . If fires occur at points uniformly chosen on localid="1646880402145" , where should the station be located so as to minimize the expected distance from the fire? That is,
choose a so as to minimize localid="1646880570154" when X is uniformly distributed over .
(b) Now suppose that the road is of infinite length鈥� stretching from point outward to . If the distance of a fire from point is exponentially distributed with rate , where should the fire station now be located? That is, we want to minimize , where X is now exponential with rate .
Consider Example 4b of Chapter 4, but now suppose that the seasonal demand is a continuous random variable having probability density function . Show that the optimal amount to stock is the value that satisfies
where is net profit per unit sale, is the net loss per unit
unsold, and is the cumulative distribution function of the
seasonal demand.
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