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Chapter 5: Q. 5.17 (page 216)
If has a hazard rate function, compute the hazard rate function of where is a positive constant.
The function's hazard rate is,.
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The median of a continuous random variable having distribution function F is that value m such that F(m) = . That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of X if X is
(a) uniformly distributed over (a, b);
(b) normal with parameters 渭,蟽;
(c) exponential with rate 位.
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.
The annual rainfall (in inches) in a certain region is normally distributed with . What is the probability that starting with this year, it will take more than years before a year occurs having a rainfall of more than inches? What assumptions are you making?
Suppose that X is a normal random variable with mean . If , approximately what is ?
The number of years that a washing machine functions is a random variable whose hazard rate function is given by
(a)What is the probability that the machine will still be working years after being purchased?
(b) If it is still working years after being purchased, what is the conditional probability that it will fail within the next
years?
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