MAT
407
Exam
2 Information
Exam
date: Thursday, November 17th
This
will be a closed notes, mostly closed
book exam.
You
will be allowed (encouraged, even) to use the tables in Appendix B of
your book. The tables that are probably
of most interest are the tables of common discrete (p. 596) and continuous (p.
597) distributions. You may also need to
use the tables for binomial and Poisson probabilities. Since there are more tables in the appendix
that you don’t need than ones that you do, please mark the pages in advance
with paper clips or something so that you don’t waste time shuffling pages.
You
will not be permitted to look through the rest of the book during the exam.
It
is expected that you will have a calculator, either a TI-83/84 or other
type. I’ll bring a few spare TI-83’s for those who want to borrow them.
3.3,
3.4
|
Discrete
distributions |
Continuous
distributions (so far) |
|
General |
General |
|
Binomial |
Uniform |
|
Geometric |
Exponential |
|
Negative
binomial |
|
|
Hypergeometric |
|
|
Poisson |
|
|
Discrete uniform |
|
Note
that I will assume that you have knowledge of material from previous sections
that were covered on the first exam in order to do some of these problems.
Office
hours: Tuesday, November 15th
12:15 – 3:00
1) On a multiple choice quiz,
there are 4 choices for the answer to each question. Suppose there are 10 questions on the
quiz. A student hasn’t studied at all,
and will answer each question purely by picking answers at random. Let X = # of questions answered correctly.
a. What
is the distribution of X?
b. What
is the expected value of the number of correct answers?
c. What
is the probability the student gets none correct?
d. What
is the probability that the student passes the quiz, that is, gets at least 6
of the 10 questions right?
2) Suppose X is uniform on [1,5].
a. Sketch the p.d.f. of X.
b. Find the c.d.f.
of X, F(x).
c. What are the mean and
variance of X?
d. Find P(X >3), P(X < 0)
and P (1< X < 3)
e. Derive the moment generating
function of X.
3) Suppose calls to a 911
switchboard in a small town occur on average 4 times in a week.
a. What would be an appropriate
model (i.e. distribution) for X = number of calls to the switchboard per
week? Justify your answer.
b. Using the model you
identified in part a., compute the probability that no calls come into the
switchboard in a week.
c. Find the probability that at
most 5 calls come in during a single week.
d. Find the probability that
more than 6 calls come in during a single week.
4) Let W = the waiting time
until the first 911 call in the last problem.
a. What is the distribution of
W?
b. What is the probability that
the first 911 call of a certain week doesn’t come in for at least 3 days? Note that 3 days = 3/7 of a week.
c. What is the expected value
of W? What is its variance?
d. Find the moment generating
function of W.
5) A coin is weighted so that P(H)=0.6. Let Y =
number of flips to get a head. What is
the probability that, if we’ve already flipped the coin 7 times without
success, that we will have to flip the coin more than 10 times to see the first
head? In other words, find P (Y > 10
| Y > 7).