Download Math 462 Exam 1 Fall 1996

Math 462 / 562
Final Exam
Fall 2013
Name: _____________________________ This is a closed book exam. You may use a calculator and the formulas
handed out with the exam. You may find that your calculator can do some of the problems. If this is so, you still need to
show how to do the problem by hand, even if you use a calculator to check your work. This is particularly true for
derivatives and integrals. On all problems, show all work and explain any reasoning which is not clear from the
computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam with your answers.
However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn in the formulas;
put them on the formula pile.
1.
You own 50 shares of stock in BigNet Corporation. Currently their stock sells at a price of $110 per share.
Furthermore this selling price is rising at a rate of $2 per day per share. However, a well placed source reports that the
SEC is investigating BigNet and eventually they will indict the President of BigNet. This will send the price of BigNet
stock plummeting to $10 per share where is will remain for the foreseeable future. You figure that from one day to the
next there is a probability of 1.5% that the announcement will come before the next day if it hasn't come already. You
want to sell your shares at a time that will maximize the expected return. Make the following assumptions and use the
following notation.
i.
Treat time as discrete. Count the days with today being day zero. Thus day n = 0 is today, day n = 1 is
tomorrow, etc.
ii. Assume the price is $110 today and that the SEC has not yet indicted the BigNet President. If the SEC has not
indicted the BigNet President on a certain day and you choose not to sell your shares on that day, then there
are two possibilities. One is that the SEC will have indicted the BigNet President before you can sell the stock
tomorrow and the price will have gone down to $10 per share where it will remain for the foreseeable future.
The other possibility is that they will not indict the BigNet President before tomorrow and you will be able to
sell the stock at tomorrow's price if you should want to. If the SEC has not indicted the BigNet President on
one day then there is a probability of 1.5% that they will do so before tomorrow and a probability of 98.5%
that they will not do so. These probabilities are independent of what has happened in previous days.
iii. Let
Sn = be the selling price of BigNet stock on day n. This is a random variable.
un = average (or expected value) of Sn.
a.
(6 points) Find the probability that the SEC will not have indicted the BigNet President before you can sell the
stock on day n should you want to.
b.
(6 points) Find a formula for un.
c.
(6 points) You want to find n so as to maximize un. If n were continuous, what equation would the optimal n
satisfy? The formula sheet might be helpful here.
d.
(5 points) Find the solution to the equation in part c.
e.
(2 points) If n is discrete, what is the optimal n?
2.
(15 points) Mary makes sandwiches at a small restaurant. The time T it takes her to make a sandwich is uniformly
distributed between one and four minutes. This means that if f(t) is the density function of T then f(t) is 1/3 for t
between 1 and 4 and f(t) is zero for t less than 1 or greater than 4. The times between which she gets a new sandwich
order are exponential distributed with mean 2 minutes. Assume all these times (both the times between new sandwich
orders and to make the sandwiches) are independent. Suppose she has just got a new sandwich order and she has just
started making this sandwich. What is the probability she will finish making this sandwich before she gets another
sandwich order?
3.
At Jack's Saloon 60% of the customers order Bud. Assume that whether one customer orders Bud is independent of
whether any other customer orders Bud.
a.
(8 points) What is the probability that exactly eight of the next ten customers order Bud?
b.
(7 points) What is the probability that exactly eight of the next ten customers order Bud and the eighth customer to
order Bud is the tenth of these next ten? Hint: First consider the next nine customers. What must be true about
them and what is the probability this occurs? Then incorporate the effect of the tenth customer.
4.
5.
Suppose X is a continuous random variable and for all x the probability that X is less than or equal to x is
a.
(5 points) Find the density function of X. The formula sheet may be helpful.
b.
(5 points) Find the probability that the value of X is between 1 and 2. For definiteness "between 1 and 2" includes
1 and 2. However, it doesn't matter since the probability of getting exactly 1 or 2 is zero since X is a continuous
random variable.
(10 points) Find the expected value of a continuous random variable X with density
 2x
f(x) =  0

6.
1
1
tan-1(x) + .
π
2
0<x<1
otherwise
Assume the times between customer arrivals at a store are independent and exponentially distributed with mean 2
minutes.
a.
(8 points) Find the probability that exactly 3 customers arrive in the first four minutes.
b.
(7 points) Find the probability that no more than 3 customers arrive in the first four minutes.
c.
(10 points) Use the central limit theorem to estimate the probability that the time it takes for the next 100
customers to arrive is between three and four hours.