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
Some Exercises From Chapter 6
Statements of the Exercises
1. Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let
Xi equal 1 if the i th ball selected is white, and let it equal 0 otherwise. Give the j
Clarication to Example in Functions of Independent Random Variables
In the document, one has two independent random variables X and Y , both of which are exponentially distributed
with parameters and . For Z D X Y , one is to nd the density function.
If f
Functions of Independent Random Variables
Let X and Y be two independent exponentially distributed random variables with parameters
min.X; Y / and Z D X Y , determine the density functions for W and Z .
and . If W D
If fX and fY are the density functions,
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
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
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
Summary of Facts Regarding Power Series
Denition of a Power Series
If x is a variable, then the innite series of the form:
an xn = a0 + a1x + a2x2 + . . . + an xn + . . .
n=0
is called a power series. More generally, the series of the form:
an (x c)n = a
Chapter 2 Exercise 15
Statement of the Exercise:
52
If it is assumed that all
5
poker hands are equally likely, what is the probability of being dealt
(a) a ush? ( A hand is daid to be a ush if all 5 cards are of the same suit.)
(b) one pair? (This occurs
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
Some Exercises from Chapter 2
Statements of the Exercises
1. Two dice are thrown. Let E be the event that the sum of the dice is odd; let F be the event
that at least one of the dice lands on 1 , and let G be the event that the sum is 5. Describe
the even
Examples of the Use of Normal Distribution
Statements of the Exercises:
1. The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally
distributed with parameters D 1:4 106 hours and D 3 105 hours. What is the
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 .