Methods of Transformations 6.32) We considered a random variable Y that has a uniform distribution on the interval [1,5]. The cost of delay is given by U=2Y 2 +3. Use the method of transformations to...

Methods of Transformations
6.32)
We considered a random variable Y that has a uniform distribution on the interval [1,5]. The cost of delay is given by U=2Y² +3. Use the method of transformations to derive the density function of U.
The method of moment-generating functions
6.42)
A type of elevator has a maximum weight capacity Y₁, which is normally distributed with mean 5000 pounds and standard deviation 300 pounds. For a certain building equipped with this type of elevator, the elevator’s load, Y₂, is a normally distributed random variable with mean 4000 pounds and standard deviation 400 pounds. For any given time that the elevator is in use find the probability that it will be overloaded, assuming that Y₁ and Y₂ are independent.
2.174)
Many public schools are implementing a “no-pass, no-play” rule for athletes. Under this system, a student who fails a course is disqualified from participating in extracurricular activities during the next grading period. Suppose that the probability is 0.15 that an athlete who has not previously been disqualified will be disqualified next term. For athletes who have been previously disqualified, the probability of disqualification next term is 0.5. If 30% of the athletes have been disqualified during the next grading period?
3.194)
One concern of a gambler is that she will go broke before achieving her first win. Suppose that she plays a game in which the probability of winning is 0.1 (and is unknown to her). It costs her $10 to play and she receives $80 for a win. If she commences with $30, what is the probability that she wins exactly once before she loses her initial capital?
3.208)
A recent survey suggests that Americans anticipate a reduction in living standards and that a steadily increasing level of consumption no longer may be as important as it was in the past. Suppose that a poll of 2000 people indicated 1373 in favor of forcing a reduction in the size of Americans automobiles by legislative means. Would you expect to observe as many as 1373 in favor of this proposition if, in fact, the general public was split 50-50 on the issue? Why?
4.164)
The length of life of oil-drilling bits depends upon the types of rock and soil that the drill encounters, but it is estimated that the mean length of life is 75 hours. An oil exploration company purchases drill bits whose length of life is approximation normally distributed with mean 75 hours and standard deviation 12 hours. What proportion of the company’s drill bits

1. Will fail before 60 hours of use?
2. Will last at least 60 hours?
3. Will have to be replaced after more than 90 hours of use?

4.126)
The weekly repair cost Y for a machine has a probability density function given by
f(y) =
with measurements in hundreds of dollars. How much money should be budgeted each week for repair costs so that the actual cost will exceed the budgeted amount only 10% of the time?
5.146)
A target for a bomb is in the center of a circle with radius of 1 mile. A bomb falls at a randomly selected point inside that circle. If the bomb destroys everything within ½ mile of its landing point, what is the probability that the target is destroyed?
5.126)
A large lot of manufactured items contains 10% with exactly one defect, 5% with more than one defect, and the remainder with no defects. Ten items are randomly selected from this lot for sale. If Y₁ denotes the number of items with one defect and Y₂, the number with more than one defect, the repair costs are Y₁+3Y₂. Find the means and variance of the repair costs.
6.100)
The time until failure of an electronic device has an exponential distribution with mean 15 months. If a random sample of five such devices are tested, what is the probability that the first failure among the five devices occurs

1. After 9 months?
2. Before 12 months?