Download Chapter 9 Additional Topics in Probability

INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 9
Additional Topics in Probability
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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
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INTRODUCTORY MATHEMATICAL ANALYSIS
9. Additional Topics in Probability
10. Limits and Continuity
11. Differentiation
12. Additional Differentiation Topics
13. Curve Sketching
14. Integration
15. Methods and Applications of Integration
16. Continuous Random Variables
17. Multivariable Calculus
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Chapter 9: Additional Topics in Probability
Chapter Objectives
• 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.
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Chapter 9: Additional Topics in Probability
Chapter Outline
9.1) Discrete Random Variables and Expected
Value
9.2) The Binomial Distribution
9.3) Markov Chains
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Chapter 9: Additional Topics in Probability
9.1 Discrete Random Variables and Expected Value
• A variable whose values depend on the outcome of
a random process is called a random variable.
Example 1 – Random Variables
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, . . .
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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
μ  μ  X   E  X    xf x 
x
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Chapter 9: Additional Topics in Probability
9.1 Discrete Random Variables and Expected Value
Example 3 – Expected Gain
An 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.
2007 Pearson Education Asia
Chapter 9: Additional Topics in Probability
9.1 Discrete Random Variables and Expected Value
Example 3 – Expected Gain
Solution:
If f is the probability function for X, then
f  179,621  P  X  179,621  0.002
f 379   P  X  379   1  0.002  0.998
The expected value of X is given by
E  X    xf x 
x
 179,621f  179,621  379f 379 
 179,6210.002  3790.998
 19
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Chapter 9: Additional Topics in Probability
9.1 Discrete Random Variables and Expected Value
Variance of X


Var  X   E  X  μ    x  μ  f x 
2
2
x
Standard Deviation of X
σ  σ  X   Var  X 
Rewriting the formula, we have
Var  X   σ 2   x 2f x   μ 2
x
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 E X   E X  
2
2
Chapter 9: Additional Topics in Probability
9.2 Binomial Distribution
• If n is a positive integer, then
a  b n n C0an  n C1an 1b  n C2an 2b2  ...  n Cn 1abn 1  n Cn bn
n
  n Ci a n i b i
i 0
Example 1 – Binomial Theorem
Use the binomial theorem to expand (q + p)4.
q  p 4  4 C0q 4  4 C1q 3 p 4 C2q 2 p2  3 C0q1p3  4 C4 p 4
4! 4 4! 3 1 4! 2 2 4! 1 3 4!

q 
q p 
q p 
qp 
p
0!4!
1!3!
2!2!
3!1!
4!0!
 q 4  4q 3 p  6q 2 p 2  4qp 3  p 4
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Chapter 9: Additional Topics in Probability
9.2 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
f x   P  X  x  n Cx p x q n  x
• The mean and standard deviation of X are given
by
  np
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  npq
Chapter 9: Additional Topics in Probability
9.2 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.
P  X  2  P  X  0   P  X  1
0
8
1
7
 1  1
 1  1
 8 C0      8 C1   
2 2
2 2
1
1 1
9
 1 1
8 

256
2 128 256
9
247

Thus, P  X  2  1 
256 256
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Chapter 9: Additional Topics in Probability
9.3 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
tij  P next state is i current state is j 
• 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
X n  TXn1
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Chapter 9: Additional Topics in Probability
9.3 Markov Chains
Example 1 – Demography
A 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:
2007 Pearson Education Asia
Chapter 9: Additional Topics in Probability
9.3 Markov Chains
Example 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.
1
2
3
Solution:
1 0.53 0.17 0.29
T 2  2  0.31 0.67 0.19
3 0.16 0.16 0.52
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.
2007 Pearson Education Asia
Chapter 9: Additional Topics in Probability
9.3 Markov Chains
Example 1 – Demography
Solution:
Initial Vector:
0.40
X0  0.30
0.30
Probability is
X3  T 3 X0  TT 2 X0
0.7 0.1 0.2 0.53 0.17 0.29 0.40 0.3368
 0.2 0.8 0.1  0.31 0.67 0.16 0.30  0.4024
 0.1 0.1 0.7 0.16 0.16 0.52 0.30 0.2608
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Chapter 9: Additional Topics in Probability
9.3 Markov Chains
Steady-State Vectors
• When T is the k × k transition matrix, the steadystate vector
 q1 
Q    
qk 
is the solution to the matrix equations
1
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 1Q  1
T  Ik Q  O