Fall, 2000

Assignments

Hints

Review for Final

Time: MWF 9:00 - MSB 303

Instructor: Professor Chuck Vinsonhaler

Office: MSB 124

Phone: 486-3944

E-mail: vinson@math.uconn.edu

Web Page: http://www.math.uconn.edu/~vinson/math231/

Office Hours: MWF 10, and by appointment

WebCT: http://webct.uconn.edu

Text: *A First Course in Probability*, Fifth Edition, by Sheldon
Ross

**Homework**: Homework will be assigned daily, and collected
about once a week. Occasionally, I will give a quiz consisting of one or
two homework problems instead of collecting homework. Homework and quizzes
count 20% of your grade. You are encouraged to work together on homework.
However, unless a problem is designated a group problem, you must turn
in your own version of the solution. Answers must be explained in full
sentences, with correct punctuation, spelling and format.

**Exams**: There will be two exams and a final. Dates will
be announced in class. Each exam counts 25% and the final 30%. You may
use a calculator and one page (8 by 11) of notes during exams.

Exam 1: Monday, October 2, 2000. Review
and Sample Exam

Exam 2: Friday, November 17, 2000 Review
and Sample Exam

**Grading**: At the end of the semester your points will be
totaled. Grades will be assigned on a fixed scale: A-90%; B-80%; C-70%;
D-60%. Plusses and minuses are assigned in the standard way. For example,
67% is a C-.

Topics:

- Combinatorial analysis permutations, combinations, multinomial coefficients
- Axioms of probability sample space, axioms, properties
- Conditional probability and independence Bayes formula
- Random variables densities, expectation, variance, Binomial, Poisson distributions
- Continuous random variables Uniform, Normal, Exponential
- Jointly distributed random variables joint distributions, sums of independent random variables.
- Properties of expectation expectation and variance of sums, conditional expectation, moment generating functions
- Limit theorems weak and strong law of large numbers, central limit theorem

1. Due September 6: p. 25 - 2,3,4,7,10,12,14,18,19,21,22,31
(may be resubmitted by 9/15)

2. Due September 13: p. 54 - 3,5,7,9,16,18,20,23,27,28,56

3. Due September 20: p. 104 - 1,2,4,5,9,13,15

4. Due September 27: p. 106 - 18,19,20,24,27,34,36,50,53

5. Due October 11: p. 173 - 14,19,20,21,23,25,37,38

6. Due October 18: p. 179 - 40,42,49,51,55,58,66,70,73,74

7. Due October 25: p. 232 - 1,5,6,7,8,11,12,14

8. Due November 1: p. 232 - 15,16,17,18,21,23,27,30,31.
A coin is flipped 10 times. Calculate the probability of at most two heads.
Then estimate this probability using the normal distribution.

9. Due November 8: p. 293 - 2,3,6,8,9,11,12,20

10. Due November 15: p. 294 - 15,17,22,23,27

11. Due November 29: p. 293 - 33,34,35,41

12. Due December 6: p. 421 - 2,3,4,5,6,7,11,14,15. Also calculate the
mgf for Uniform(0,1).

**Extra Credit Problems**: These are problems to be carefully written
up and submitted before the last class of the semester for extra credit.

EC 1: p.19 - #32

EC 2: A 3-way argument breaks out in Probability about grades on an
exam and the participants decide to settle the matter with a dual. They
will draw straws to see who shoots first, second and third. Then they will
stand on the vertices of an equilateral triangle and fire single shots
in order, continuing the cycle until only one person is left standing.
The shooter may aim wherever (s)he chooses. Alexi is known to be accurate
100% of the time, while Brenda is 75% accurate and Chuck 60% accurate.
If Chuck draws the first shot he will miss intentionally because Alexi
will surely fire at Brenda. Calculate the probabilities of survival for
each of the contestants. There are 6 different cases and you will need
to use geometric series.

EC 3: A fair coin is flipped repeatedly. Which sequence is more likely
to appear first, HTHT or HTHH? Which sequence is more likely to appear
most often?

Math 231Q-01

Review for Exam 1

Fall, 2000

Basic Counting Principles

? Count the choices at each stage and multiply

? Count the choices for each case and add

Arranging (Permutations): Draw a picture ___ ___ ___

Choosing (Combinations): Use binomial coefficients

Sample Spaces and Events

? Axioms

? Venn Diagrams

? Equally Likely Outcomes: P(E) =

Conditional Probability

? Bayes Theorem (calculate the probability in each case and add)

? Multiplication Rule (calculate the probability at each stage
and multiply)

? Independence

Random Variables

? Cumulative Distribution Function (cdf): F(b)

? Probability Mass Function (pmf): p(x)

**Sample Exam 1 Problems.**

1. How many different letter arrangements can be made using (all)
the letters in SCIENCE?

2. In a certain club, 45% enjoy red wine, 35% enjoy white and
20% enjoy both. What percentage enjoy at least one of the two types of
wine?

3. Urn A contains 3 white and 1 black ball; urn B contains 3
white and 5 black; urn C contains 3 yellow and 1 white ball. An urn is
chosen at random and a ball drawn.

A.What is the probability the ball is white?

B. Given the ball is white, what is the probability
urn C was chosen?

C. Let X be the random variable that is 0 if a white
ball is drawn, 1 if a black ball is drawn and 2 if a yellow ball is drawn.
Describe the cmf and pmf for X.

4. There is a 50% chance that the Queen carries the gene for
hemophilia. If she is a carrier then each prince (male child) has a 50%
chance of having hemophilia. Suppose the Queen has 3 princes and none have
the disease.

A. What is the probability the queen is a
carrier?

B. If the fourth child is a prince, what is
the probability he will have homophilia.

C. Let E be the event that the first 3 princes
do not have hemophilia and F the event that the fourth has no hemophilia.
Are E and F independent?

Answers:

1. 1260

2. 60%

3A. 11/24

3B. 2/11

3C. The cmf for X has 3 "jumps." Where and what size are they?

4A. 1/9

4B. 1/18

4C. First, what do you think? We want to show that the probability
of F given E is the same as the probability of F (if we can).

Here is a more detailed solution of Problem 4.

E: first three princes do not have hemophilia

F: fourth does not have hemophilia

Q: Queen carries the gene.

4A. P(Q given E) = P(E and Q)/P(E)

P(E) = P(E and Q) + P(E and Q complement) = (½)(½)(½)(½)
+ (½)1 = 9/16.

Therefore P(Q given E) = 1/9

4B. P(F given E) = P(F and E)/P(E)

P(F and E) = P(F and E and Q) + P(F and E and Q complement) = (½)(½)(½)(½)(½)
+ 1/2 = 17/32

Therefore P(F given E) = (17/32)/(9/16) = 17/18. Thus, P(F complement,
given E) = 1/18.

4C. P(F) = P(F and Q) + P(F and Q complement) = (½)(½)
+ ½ = 3/4. Since P(F) is not equal to P(F given E), the events E
and F are dependent. This should be inituitively obvious. If hemophilia
has not shown up in the first three princes, it indicates the Queen is
not a carrier so the fourth prince will not have hemophilia.

**Review for Exam 2**

Discrete distributions

probability mass function, cumulative distribution
function

Expected Value, Variance

Binomial (n,p)

Poisson (lambda) (including approximation of Binomial,
lambda = np)

Continuous Distributions

probability density function, cumulative distribution
function

Expected Value, Variance

Uniform(a,b), Normal(mu, sigma squared), Exponential(lambda)

Normal approximation of Binomial, mu = np, sigma
squared = np(1-p)

Use of Normal(0,1) table

Joint Distributions

F(x,y), p(x,y), f(x,y)

Marginal distributions

Independence

Sums of random variables

Suggested Review Problems: p. 293 - 2,3,9,20;
p. 232 - 1,6,11,15,18,23,30 ;

p. 173 - 14,19,38,40,55,58,73

Sample Exam 2.

1. Suppose on average there is one homicide a day in Manhattan.
What is the probability that next week there will be fewer than 2 homicides
in Manhattan?

2. Suppose the number of hours people sleep in a night is normally
distributed with mean 8 and standard deviation 1. One hundred people are
selected at random. What is the approximate probability that more than
20 of these people will sleep more than 9 hours tonight?

3. If the continuous random variable X has pdf f(x) = cx(1-x), 0<x<1
and is 0 elsewhere, find E[X] and Var[X].

4. Suppose X is a normal random variable with mean m and variance
25. Suppose that P(X>c)=0.1 and P(X>2c)=0.05. Find m and c.

5. What is the approximate probability that if you toss a fair
coin 200 times, the number of heads is strictly larger than 98 but smaller
than 102?

6. Suppose X and Y have a joint pdf given by f(x,y) = 3x, 0<y<x,
0<x<1.

(a) Are X and Y independent?

(b) Find E[Y].

(c) Find P(X>1/2 given that Y<3/4).

(d) Find P(X>2Y)

7. Solve Buffon's Needle Problem, or some variant of it.

Answers to Sample Exam 2. (*My answers - check them*!)

1. 8exp(-7)

2. .1292

3. 1/2,1/20

4. -4.595, 1.815

5. .1664

6. (a) no

(b) 3/8

(c) under construction

(d) 1/2

7. See page 255 in the text

Review Session: Monday, December 18, 3:30 PM, MSB 215

**Key Concepts**: Counting Principles, Equally likely
outcomes, pmf, pdf, cdf, expected value, variance, conditional probability,
independence, approximating binomial distribution with the normal, joint
distributions,
moment generating function, Markov inequality, Chebyshev inequality, Weak
Law of Large Numbers, Central Limit Theorem.

1. Suppose X is a Poisson random variable with parameter 3 and Y is a Poisson random variable with parameter 5. If X and Y are independent, what is P(XY = 4)?

2. Suppose X1, ,X25 are independent and identically distributed random variables with mean 1 and standard deviation s. If P(X1 + + X25 >100) = 0.2, give an estimate of s.

3. A drug is tried on a group of 100 people with AIDS. These people were found to live an average of 1 year longer than expected, with a standard deviation of 4 years. Suppose we let Xi represent the increase in expected lifetime of the ith person, and assume that the Xi are i.i.d with mean 0 and standard deviation 4. If Y is the average of the Xi for i = 1,2, ,100, what is the probability of the observed result, that is, Y > 1?

4. A person is dealt 3 cards from a standard 52 card deck. What is the probability that he will have exactly two different suits?

5. Two random variables X and Y have joint density f(x,y) = 8xy for 0< x<y and 0<y<1. Are X and Y independent? What is the probability X+Y < 1?

6. If the continuous random variable X has density f(x) = cx^(-5) for x > 1 and 0 otherwise, find E(X) and Var(X).

7. What is the approximate probability that if you roll a fair die 60 times you will get a 3 exactly 10 times? Use the normal approximation.

8. A fair die is rolled. If X is the number that comes up, what is the moment generating function for X?

9. Write a paragraph discussing the difference between the Weak Law of Large Numbers and the Central Limit Theorem.

Answers by C. Vinsonhaler

1. 605[exp(-8)]/4

2. s = 17.8

3. 0.0062

4. 234/425

5 (a) No

(b) 1/6

6. E(X) = 4/3, Var(X) = 2/9

7. 0.14

8. M(t) = [exp(t)+exp(2t)+exp(3t)+exp(4t)+exp(5t)+exp(6t)]/6

9. See the text and class notes