FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Solutions to Assignment 3
1.
2.
The probability mass function of X is given by
4
4 2
2
11 8
6
P X 4
, P X 2 ,
91
14 91
14
2
2
48
1 1 32
P X 1 , ,
91
14
2
2
2
1
P X

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Assignment 6
No need to hand in the solutions.
1.
You are given the following information about N, the annual number of claims for a randomly
selected insured:
1
1
1
P N 0 , P N 1

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 1
10 12
1. There are possible choices of the 5 men and 5 women. They can be paired up in 5!
5 5
ways, since if we arbitrarily order the men then the firs

Introduction
FINA2220A
Quantitative Methods for
Actuarial Analysis
In this chapter, we introduce the concept of the probability of
an event
Then we show how these probabilities can be computed in
certain situations
As a preliminary, we need the concept

Joint Distribution Functions
There
are probability statements concerning two or more
random variables
In order to deal with such probability, we define the joint
cumulative probability distribution function
For any two random variables X and Y, the joi

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 2
1. Let E be the event that a randomly chosen pregnant women has an ectopic pregnancy and S the
event that the chosen person is a smoker. Then the problem

Introduction
In this chapter we develop and exploit additional properties of
expected values
To begin, recall that the expected value of the random variable X is
defined by
FINA2220A
Quantitative Methods for
Actuarial Analysis
where X is a discrete random

Conditional Variance
We can define the conditional variance of X given that Y = y, which is
defined as follows:
FINA2220A
Quantitative Methods for
Actuarial Analysis
That is, varX(X | Y) is equal to the (conditional) expected square of the
difference betw

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 4
1.
The probability density function f(x) = k(10 + x)2. Hence we have
1 0 k 10 x dx k 110 x
2
40
1 40
0
1 1 2k
k
10 50 25
P X 5 0 12.5 10 x dx 12.5 11

Introduction
A
typical problem of interest involving probability
A communication system is to consist of n seemingly identical
antennas that are to be lined up in a linear order
The resulting system will be called functional as long as no two
consecutiv

Random Variables
When an experiment is performed, we are mainly interested in
FINA2220A
Quantitative Methods
for Actuarial Analysis
Chapter 4
Random Variables
some function of the outcome as opposed to the actual outcome
itself
Examples:
The sum of the

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 3
1.
2.
The probability mass function of X is given by
4
4 2
2
11 8
6
P X 4
, P X 2 ,
91
14 91
14
2
2
48
1 1 32
P X 1 , ,
91
14
2
2
2
1
P X

x1 Disuihuon
The table below gives the value x3 for which P[x2 < x: ] = P for a given number of
degrees of freedom and a given value of P.
Values of P
Freedom 0.005 0.010 7.025 0.050 0.100 0.900 0.950
U": :_ - 0.001 . 2.706 3.841
0.484
0.831

Introduction
An important concept Conditional Probability
The importance of this concept is twofold:
Calculating probabilities when some partial information
concerning the result of the experiment is available
Desired probabilities are conditional
Com

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 6
1.
The conditional probability density (mass) function of S given N is
P S 0 N 0 1,
f S N s 1 0.2e 0.2 s , s 0,
f S N s 2 0.125e 0.125 s , s 0.
Hence the

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Solutions to Assignment 5
1.
Let Wi be the events that the ith ball selected is white and Ri be the events that the ith ball
selected is red.
(a) Let p(x1, x2) be the joint probabi

Introduction
The most important theoretical results in probability theory are
limit theorems
Of these, the most important are classified under
Laws of large numbers
FINA2220A
Quantitative Methods
for Actuarial Analysis
Stating conditions under which th

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Solutions to Assignment 6
1.
The conditional probability density (mass) function of S given N is
P S 0 N 0 1,
f S N s 1 0.2e 0.2 s , s 0,
f S N s 2 0.125e 0.125 s , s 0.
Hence the

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Assignment 3
Hand in the solutions on or before 26 October 2015.
1.
Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls.
Suppose that we win $

Introduction
In this chapter we develop and exploit additional properties of
expected values
To begin, recall that the expected value of the random variable X is
defined by
FINA2220A
Quantitative Methods for
Actuarial Analysis
where X is a discrete random

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Assignment 6
No need to hand in the solutions.
1.
You are given the following information about N, the annual number of claims for a randomly
selected insured:
1
1
1
P N 0 , P N 1

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Assignment 5
Hand in the solutions on or before 27 November 2015.
1.
Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8
red balls. Let Xi e

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Solutions to Assignment 5
1.
Let Wi be the events that the ith ball selected is white and Ri be the events that the ith ball
selected is red.
(a) Let p(x1, x2) be the joint probabi

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Assignment 4
Hand in the solutions on or before 4 November 2015.
1.
The lifetime of a machine part has a continuous distribution on the interval (0, 40) with
probability density fu

Joint Distribution Functions
There
are probability statements concerning two or more
random variables
In order to deal with such probability, we define the joint
cumulative probability distribution function
For any two random variables X and Y, the joi

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2015-2016
Solutions to Assignment 4
1.
The probability density function f(x) = k(10 + x)2. Hence we have
1 0 k 10 x dx k 110 x
2
40
1 40
0
1 1 2k
k
10 50 25
P X 5 0 12.5 10 x dx 12.5 11

Introduction
The most important theoretical results in probability theory are
limit theorems
Of these, the most important are classified under
Laws of large numbers
FINA2220A
Quantitative Methods
for Actuarial Analysis
Stating conditions under which th

FINA2220A Quantitative Methods for Actuarial Analysis
First Term 2016-2017
Monday 12:30pm 2:15pm, Room 407, Wu Ho Man Yuen Building
Wednesday 2:30pm 4:15pm, Room G04, An Integrated Teaching Building
Instructor: Professor Albert C.S. Wong
Office: Room 121

Study Manual for
Exam P/Exam 1
Probability
16-th Edition
by
Dr. Krzysztof Ostaszewski
FSA, CERA, FSAS, CFA, MAAA
Note:
Note:
NO
RETURN
NO RETURN IFIF OPENED
OPENED
TO OUR READERS:
Please check A.S.M.s web site at www.studymanuals.com for errata and
update

Bernoulli
Overview
X:0or1
1) Singleflipofacoinresultinginheadortail
2) Launchofaproductresultingineithersuccessorfailure
Parameters
p:probabilityofX=1
PDF/PMF
P X
1
p, P X
0
1
p
q
CDF
P X
x
1
0
0
p
1
E(X)
E X
p
Var(X)
Var X
pq
M t
MGF
KeyPoints
q
pe
x
x

Introduction
Apart
from discrete random variables, there also exist random
variables whose set of possible values is uncountable
Life time of a transistor
FINA2220A
The
time that a train arrives at a specified stop
We say that X is a continuous random

SOCIETY OF ACTUARIES
EXAM P PROBABILITY
EXAM P SAMPLE SOLUTIONS
Copyright 2016 by the Society of Actuaries
Some of the questions in this study note are taken from past examinations.
Some of the questions have been reformatted from previous versions of thi