Exercise 12 p.248
Statement:
A bus travels between the two cities A and B , which are 100 miles apart. If the bus has a breakdown, the distance
from the breakdown to city A has a uniform distribution over . 0; 100 /. There is a bus service in city A, in B
Chapter 1: Combinatorial Analysis
The mathematical theory of counting is formally known as combinatorial analysis.
Basic Principle of Counting
Suppose that two experiments are to be performed. Then if experiment 1 can result in any one of m possible
outco
Chapter 3: Conditional Probability and Independence
Section 1: Introduction
Conditional probability is a concept used in
1. calculating probabilities when some partial information concerning the result of the experiment is
concerned;
2. computing desired
Chapter 2: Axioms of Probability
Section 2: Sample Space and Events
For a given experiment, the set of all possible outcomes is called the sample space of the experiment and is
denoted by S . Any subset E of a sample space is known as an event. If the out
Chapter 8: Limit Theorems
Section 1: Introduction
The most important theoretical results in probability theory are the limit theorems. Of these theorems, the ones of
most value are either laws of large numbers or central limit theorems. The theorems consi
Chapter 4: Random Variables
Section 1: Random Variables
Definition: If S is a sample space for an experiment, then a random variable X is a real-valued function of S ,
i.e., X : S 6 R .
If X is a random variable of S and there is a probability given for S
Chapter 7: Properties of Expectation
Section 1: Introduction
Definition: If X is a discrete random variable with probability mass function p(x) , then the expectation or the
expected value of X , denoted by E[ X ] , is defined by E[ X ] =
.
If X is a cont
Chapter 5: Continuous Random Variables
Section 1: Introduction
Definition: X is called a continuous random variable if there exists a non-negative function f , defined for all real
x ( -4, 4 ) , having the property that for any set B of real numbers, P( X
Chapter 6: Jointly Distributed Random Variables
Section 1: Joint Distribution Functions
Definition: For any two random variables X and Y , the joint cumulative distribution function of X and Y is
defined by F(a, b) = P( X # a , Y # b ) , -4 < a , b < 4 .
Math 230 Worksheet #4
February 9, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
Math 230 Worksheet #3
January 26, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
Math 230 Worksheet #5
February 16, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
Math 230 Worksheet #6
March 1, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
tog
Math 230 Worksheet #2
January 19, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
Math 230 Worksheet #1
January 12, 2012
NetID:
Partner(s):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn in your worksheet
Math 230 Worksheet #3
January 26, 2012
NetID: l H 001 I Partner(s): l L- 9? 3/
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Tu
Math 230 Worksheet #6
March 1, 2012
NetID: Partner (5):
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your coliaborator on your worksheet. Turn in your worksheet
to
Math 230 Worksheet #1
January 12, 2012
NetID: 016361000 Partner(s): ZZZ a? l
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your worksheet. Turn
SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY
EXAM P PROBABILITY
EXAM P SAMPLE SOLUTIONS
Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society
Some of the questions in this study note are taken from past SOA/CAS examinations.
P-0
Exercise 15, Page 115
Statement:
One probability class of 30 students contains 15 that are good, 10 that are fair, and 5 that are of poor quality. A
second probability class, also of 30 students, contains 5 that are good, 10 that are fair, and 15 that are
Solution to Exercise #14 on p.188
Statement of Exercise :
Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players
compare their numbers, the one with the higher one is declared the winner. Initially, players 1
An Example of Mathmatical Finance
This example is to illustrate the amount of knowledge necessary to address a relatively straightforward
application of probability in the eld of mathematical nance.
Example:
Starting at some xed time, let S (n) denote the
Exercise 41 p.251
Statement :
Find the distribution of R D A sin. /, where A is a xed constant and
is uniformly distributed on .
=2; =2/.
Solution:
There are two methods for determining the distribution of R: using the formula given on p.243 or using the
Some Exercises from Chapter 1
Statements of the Exercises
1. Twenty workers are to be assigned to 20 dierent jobs, one to each job. How many dierent
assignments are possible?
2. John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If e
Chapter 4 Example 4b, p.129
Statement:
A product that is sold seasonally yields a net prot of b dollars for each unit sold and a net loss of l dollars for
each unit left unsold when the season ends. The number fo untis of the product that are ordered at a
Solution to Exercise #23 on p.190
Statement:
You have $1000 and a certain commodity presently sells for $2 per ounce. Suppose that after one week the
dommodity will sell for either $1 or $4 per ounce, with these two possibilities being equally likely.
(a)
Some Exercises from Chapter 2 Exercise 5(c)
In this part of the problem, the sample space S consists of ordered pairs of students. Thus all events consists of
such pairs.
Let S t 1 be the event that the rst of the two students is taking a language class a
Exercises from Chapter 4
Statements of the Exercises
1. Two fair dice are rolled. Let X equal the product of the 2 dice. Compute P (X = i) for i = 1, 2, . . .
2. Five men and ve women are ranked according to their scores on an examination. Assume that
no
Some Exercises from Chapter 3
Statements of the Exercises
1. If two fair dice are rolled, what is the conditional probability that the rst one lands on 6 given
that the sum of the dice is i? Compare all the values of i between 2 and 12.
2. An urn contains
Some Exercises From Chapter 5
Statements of the Exercises
1. A system consists of one original unit plus one spare can function for a random amount of time
X . If the density of X is given (in units of months) by
f (x) =
C xex/2
0
x>0
x0
what is the proba
75
Math 230 Worksheet #2
January 19, 2012
NetID: (A b5? l2 3 Partner(s): 5196i 5 Z I
Instructions: You may work with a partner on this worksheet, but you must write you own solutions
to the problems and write the NetID of your collaborator on your workshe