Department of Mathematics and Statistics STAT 3083 - UNB ...

Department of Mathematics and Statistics
STAT 3083
Assignment 3 Due: Friday, 6 October, 2006.
1. A rental agency which leases heavy equipment by the day has found that one particular
(expensive) piece of equipment is leased, on average, one day in five. If rental on one
day is independent of rental on any other day, find the probability distribution of Y
which is equal to the number of days between a pair of rentals.
2. Let Y be a random variable with probability function given by the following table:
y 1 2 3 4
P (Y = y) 0.4 0.3 0.2 0.1 Find E(Y ), E(1/Y ), E(Y 2 − 1), and V(Y )
3. Problem 58, page 92 of Milton and Arnold.
4. The bits for drills called “Kelley bars” used in blasting soft rock such as limestone
must be changed fairly often. Let X be the number of holes that can be drilled per
bit. Suppose that it is known that the probability function f(x) for X is given by
x 1 2 3 4 5 6 7 8
f(x) 0.02 0.03 0.05 0.20 0.40 0.20 0.07
(a) Find f(8). (Note that the foregoing table implies that 1, 2, . . . , 8 are the
only possible values of X.)
(b) Construct the table for the cumulative distribution function F (x) for X.
(c) Use F to find the probability that a randomly selected bit can be used to drill
between 3 and 5 (inclusive) holes.
(d) Find F (3), F (−3), F (4.267) and F (10.1).
(e) Find P (X ≤ 4) and P (X < 4).
(f) Find µ = E(X).
(g) Find σ 2 = V(X).
(h) Find E(2 X ).
5. Suppose that 35% of all printers used on home computers require some adjustment
at the time of installation. The remainder operate correctly. A particular dealer sells
42 units during a particular month.
(a) Let X be the number of units which operate correctly upon installation. Find the
mean and standard deviation of X.
. . . . . . (continued over page)

(b) Use Minitab to find
(i) the probability that exactly 30 of the units operate correctly upon installation.
(ii) the probability that at least 30 of the units operate correctly upon installation.
(iii) the probability that at most 30 of the units operate correctly upon installation.
(Click on
Calc --> Probability Distributions --> Binomial
and then fill in the appropriate bits in the menu that appears. Note that you will
want to click on “Input constant”; you use “Input column” when you want
to calculate probability functions at a number of “x” values and you have these
stored in a column of the Minitab worksheet — you then “input” the designation
of that column.)
6. A feeder airline overbooks flights on small jet-prop aircraft, which seats 20 passengers,
by 5 passengers. (I.e. they book 25 people on the 20 seat aircraft.) It is known that
on average only 80% of booked passengers actually arrive for any given flight and
claim their seats. If more than 20 booked passengers arrive for a flight the airline has
to pay compensation to those who cannot get on the flight. Suppose that the airline
has to pay $175 compensation to any passenger who cannot get on the flight. Let X
be the amount of compensation that the airline has to pay with respect to a single
flight.
(a) Make a table which specifies the probability function for the random variable X
(i.e. tabulate the possible values for X, and the probability that X takes on each
of these values.) (Use Minitab to perform the associated calculations; click on
“Calc”, then on “Probability Distributions”, etc.)
(b) Calculate the expected value of X.
(c) Calculate the standard deviation of X.
7. Problem 42, page 89 of Milton and Arnold. (You can save yourself some tedious
arithmetic by making use of the tables in Appendix A of your text.)
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