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Chapter 5: Q. 5.22 (page 216)
Compute the hazard rate function of a gamma random variable with parameters and show it is increasing when and decreasing when
That is, implies when .
is increasing when similarly, is decreasing when
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Let be a random variable that takes on values betweenand. That is.Show that
Hint: One approach is to first argue that
localid="1646883602992"
and then use this inequality to show that
A fire station is to be located along a road of length. If fires occur at points uniformly chosen on, where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to
minimize
whenis uniformly distributed over
Now suppose that the road is of infinite length鈥� stretching from point outward to. If the distance of fire from the point is exponentially distributed with rate, where should the fire station now be located? That is, we want to minimize, where is now exponential with rate.Verify thatwhenis a gamma random variable with parameters and
The lung cancer hazard rateof a -year-old male smoker is such that
Assuming that a -year-old male smoker survives all other hazards, what is the probability that he survives to
ageand age 60 without contracting lung cancer?
The number of years a radio function is exponentially distributed with the parameterIf Jones buys a used radio, what is the probability that it will be working after an additionalyear?
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