Problems 197
(Hint: In order forYto equalk, how many successes must result in the firstk− 1
trials and what must be the outcome of trialk?)
(d) Show that
E[Y]=r/p
(Hint: WriteY=Y 1 +...+YrwhereYiis the number of trials needed to go
from a total ofi−1 to a total ofisuccesses.)
21.IfUis uniformly distributed on (0, 1), show thata+(b−a)U is uniform
on (a,b).
22.You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some
time uniformly distributed between 10 and 10:30. What is the probability that
you will have to wait longer than 10 minutes? If at 10:15 the bus has not yet
arrived, what is the probability that you will have to wait at least an additional
10 minutes?
23.IfXis a normal random variable with parametersμ=10,σ^2 =36, compute
(a)P{X> 5 };
(b) P{ 4 <X< 16 };
(c) P{X< 8 };
(d) P{X< 20 };
(e)P{X> 16 }.
24.The Scholastic Aptitude Test mathematics test scores across the population of
high school seniors follow a normal distribution with mean 500 and standard
deviation 100. If five seniors are randomly chosen, find the probability that
(a)all scored below 600 and(b)exactly three of them scored above 640.
25.The annual rainfall (in inches) in a certain region is normally distributed with
μ=40,σ=4. What is the probability that in 2 of the next 4 years the rainfall
will exceed 50 inches? Assume that the rainfalls in different years are independent.
26.The width of a slot of a duralumin forging is (in inches) normally distributed with
μ=.9000 andσ=.0030. The specification limits were given as .9000±.0050.
What percentage of forgings will be defective? What is the maximum allowable
value ofσthat will permit no more than 1 in 100 defectives when the widths are
normally distributed withμ=.9000 andσ=.0030?
27.A certain type of lightbulb has an output that is normally distributed with mean
2,000 end foot candles and standard deviation 85 end foot candles. Determine
a lower specification limitLso that only 5 percent of the lightbulbs produced
will be defective. (That is, determineLso thatP{X≥L}=.95, whereXis the
output of a bulb.)
28.A manufacturer produces bolts that are specified to be between 1.19 and
1.21 inches in diameter. If its production process results in a bolt’s diameter