Stat 231 Exam I
October 6, 1997
Professor Vardeman
1. The random variable \ œ the number of hours till failure of a WD disk drive
is described using an exponential distribution with mean 15,000 hours.
(a) Evaluate the probability that a given drive lasts at least 20,000 hours.
(b) A new computer network has 10 of these drives installed on computers in the network. Use
your answer to (a) and an assumption of independence of the 10 drive lifetimes and evaluate the
probability that at least 9 of these drives are failure-free through 20,000 hours. No need to
simplify. (If you could not do (a) use the incorrect value of .17.)
(c) A WD warranty program pays customers for early failure of drives of this type according to
the following schedule:
Failure in the first 10,000 hours produces a payment of $300
B
Failure at B hours (10,000  B  20,000) produces a payment of $300Ð2  "!ß!!!
Ñ
Failure at B hours with B  20,000 produces no payment
Set up (but do not attempt to evaluate) an expression for the mean warranty payment under this
plan. (This is E2Ð\Ñ for an appropriate function 2.)
-1-
2. There are three possible classifications of glass vials of a certain type, namely conforming,
blemished (but useable) and defective. Two large lots of these vials have the following
compositions
Lot 1: 70% conforming, 20% blemished and 10% defective
Lot 2: 80% conforming, 10% blemished and 10% defective
Lot 1 is three times the size of Lot 2 and these two lots have been mixed in a storeroom.
Suppose that I go to the storeroom and select a vial at random to use in a chemical analysis.
(a) What is the probability that the vial is from Lot 1 and not defective?
(b) What is the probability that the vial is blemished?
(c) What is the conditional probability that the vial is from Lot 1 given that it is blemished?
(d) Use your answer to (b) and evaluate the mean number of vials I will have to draw from the
storeroom in order to find a blemished vial.
-2-
3. Fill levels of nominally 20 oz. glasses of cola filled by a vending machine are approximately
normal with mean . œ 20.25 oz and standard deviation 5 œ .2 oz.
(a) About what fraction of glasses are filled below the nominal level by this machine?
(b) What filling precision (as measured by 5) would be required in order to keep the mean
amount of cola given away to the level provided by the current machine (.25 oz per glass) but
ensure that only 1% of glasses are underfilled?
4. Suppose that \" and \# are independent random variables with the same mean, ., and
respective variances 5"# and 5## . The sample variance W # of these two variables is also a random
variable. Find its mean, EW # . (Hint: You may use the fact that
#
" !
#Ñ
W # œ #"
Ð\3  \Ñ# œ Ð\" \
)
#
-3-
5. Two random variables \ and ] have a joint probability function 0 ÐBß CÑ given in the table
below.
B
1
2
3
4
6 .1 .05 .05 0
.1 .05
C 5 .05 .1
4 .05 .1
.1 .05
3 0 .05 .05 .1
(a) Give the marginal probability function for the variable ] , 2ÐCÑ.
(b) Use your answer to (a) and find Var ] . No need to simplify.
(c) Let L be the cumulative probability function for ] . Evaluate LÐ4.15Ñ.
(d) Are the random variables \ and ] independent? (Convince me one way or the other.)
-4-
6. A student decides to use the random digit function on her calculator to select a three digit PIN
number for use with her new ATM card. (Assume that all numbers 000 through 999 are then
equally likely to be chosen.)
(a) What is the probability that her number uses only odd digits?
(b) What is the probability that all three digits in her number are different?
(c) What is the probability that her number uses three different digits and lists them in either
ascending or descending order?
-5-
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