chapter 9 - additional topics in probability

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 9 Chapter 9 Additional Topics in ProbabilityAdditional Topics in Probability


INTRODUCTORY MATHEMATICAL ANALYSIS

0. Review of Algebra

1. Applications and More Algebra

2. Functions and Graphs

3. Lines, Parabolas, and Systems

4. Exponential and Logarithmic Functions

5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability10. Limits and Continuity11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. Multivariable Calculus

INTRODUCTORY MATHEMATICAL ANALYSIS


• To develop the probability distribution of a random variable.

• To develop the binomial distribution and relate it to the binomial theorem.

• To develop the notions of a Markov chain and the associated transition matrix.

Chapter 9: Additional Topics in Probability

Chapter ObjectivesChapter Objectives


Discrete Random Variables and Expected Value

The Binomial Distribution

Markov Chains

9.1)

9.2)

9.3)


Chapter OutlineChapter Outline



9.1 Discrete Random Variables and Expected Value9.1 Discrete Random Variables and Expected Value

Example 1 – Random Variables

• A variable whose values depend on the outcome of a random process is called a random variable.

a.Suppose a die is rolled and X is the number that turns up. Then X is a random variable and X = 1, 2, 3, 4, 5, 6.

b. Suppose a coin is successively tossed until a head appears. If Y is the number of such tosses, then Y is a random variable and Y = y where y = 1, 2, 3, 4, . . .


Chapter 9: Additional Topics in Probability 9.1 Discrete Random Variables and Expected Value

Example 1 – Random Variables

c. A student is taking an exam with a one-hour limit. If X is the number of minutes it takes to complete the exam, then X is a random variable. Values that X may assume = (0,60] or 0 X 60.

• If X is a discrete random variable with distribution f, then the mean of X is given by

x

xxfXEXμμ


Chapter 9: Additional Topics in Probability9.1 Discrete Random Variables and Expected Value

Example 3 – Expected GainAn insurance company offers a $180,000 catastrophic fire insurance policy to homeowners of a certain type of house. The policy provides protection in the event that such a house is totally destroyed by fire in a one-year period. The company has determined that the probability of such an event is 0.002. If the annual policy premium is $379, find the expected gain per policy for the company.


Chapter 9: Additional Topics in Probability9.1 Discrete Random Variables and Expected ValueExample 3 – Expected Gain

Solution: If f is the probability function for X, then

The expected value of X is given by

998.0002.01379379

002.0621,179621,179

XPfXPf

19998.0379002.0621,179

379379621,179621,179

ff

xxfXEx


Chapter 9: Additional Topics in Probability9.1 Discrete Random Variables and Expected Value

Variance of X

Standard Deviation of X

Rewriting the formula, we have

xfμxμXEXVarx 22

XVarXσσ

22222 XEXEμxfxσXVarx



9.2 Binomial Distribution9.2 Binomial Distribution

Example 1 – Binomial Theorem

• If n is a positive integer, then

Use the binomial theorem to expand (q + p)4.

iin

n

iin

nnn

nnn

nn

nn

nn

n

baC

bCabCbaCbaCaCba

0

11

222

110

...

432234

3122134

444

3103

2224

314

404

4

464 !0!4

!4!1!3!4

!2!2!4

!3!1!4

!4!0!4

pqppqpqq

ppqpqpqq

pCpqCpqCpqCqCpq


Chapter 9: Additional Topics in Probability9.2 Binomial Distribution

Binomial Distribution• If X is the number of successes in n independent

trials, probability of success = p and probability of failure = q, the distribution f for X is

• The mean and standard deviation of X are given by

xnxxn qpCxXPxf

npqnp


Chapter 9: Additional Topics in Probability9.2 Binomial Distribution

Example 3 – At Least Two Heads in Eight Coin Tosses

A fair coin is tossed eight times. Find the probability of getting at least two heads.Solution:X has a binomial distribution with n = 8, p = 1/2, q = 1/2.

Thus,

2569

1281

218

256111

21

21

21

21

10271

18

80

08

CC

XPXPXP

256247

256912 XP



9.3 Markov Chains9.3 Markov Chains• A Markov chain is a sequence of trials in which

the possible outcomes of each trial remain same, are finite in number, and have probabilities dependent upon the outcome of the previous trial.

• The transition matrix for a k-state Markov chain is

• State vector Xn is a k-entry column vector in which xj is the probability of being in state j after the nth trial.

• T is the transition matrix and Xn is given by

jiPt ij is state current is state next

1 nn TXX


Chapter 9: Additional Topics in Probability9.3 Markov Chains

Example 1 – DemographyA county is divided into 3 regions. Each year, 20% of the residents in region 1 move to region 2 and 10% move to region 3. Of the residents in region 2,10% move to region 1 and 10% move to region 3. Of the residents in region 3, 20% move to region 1 and 10% move to region 2.a. Find the transition matrix T for this situation. Solution:


Chapter 9: Additional Topics in Probability9.3 Markov ChainsExample 1 – Demography

b. Find the probability that a resident of region 1 this year is a resident of region 1 next year; in two years.

Solution:

c. This year, suppose 40% of county residents live in region 1, 30% live in region 2, and 30% live in region 3. Find the probability that a resident of the county lives in region 2 after three years.

52.016.016.019.067.031.029.017.053.0

321

T

3 2 1

2


Chapter 9: Additional Topics in Probability9.3 Markov ChainsExample 1 – Demography

Solution: Initial Vector:

Probability is

30.030.040.0

X0

2608.04024.03368.0

30.030.040.0

52.016.016.016.067.031.029.017.053.0

7.01.01.01.08.02.02.01.07.0

XTTXTX 02

03

3


Chapter 9: Additional Topics in Probability9.3 Markov Chains

Steady-State Vectors• When T is the k × k transition matrix, the steady-

state vector

is the solution to the matrix equations

kq

qQ

1

OQIT

Q

k 111