contemporary engineering economics, 4 th edition, © 2007 probabilistic cash flow analysis lecture...
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Contemporary Engineering
Economics, 4th edition, © 2007
Probabilistic Cash Flow Analysis
Lecture No. 47Chapter 12Contemporary Engineering EconomicsCopyright, © 2006
Contemporary Engineering
Economics, 4th edition, © 2007
Probability Concepts for Investment Decisions Random variable: variable that
can have more than one possible value
Discrete random variables: random variables that take on only isolated (countable) values
Continuous random variables: random variables that can have any value in a certain interval
Probability distribution: the assessment of probability for each random event
Contemporary Engineering
Economics, 4th edition, © 2007
Types of Probability Distribution Continuous Probability Distribution
Triangular distribution Uniform distribution Normal distribution
Discrete Probability Distribution
Contemporary Engineering
Economics, 4th edition, © 2007
F x P X x p jj
j
( ) ( )
1
(for a discrete random variable)
(for a continuous random variable) f(x)dx
Cumulative Probability Distribution
Contemporary Engineering
Economics, 4th edition, © 2007
Useful Continuous Probability Distributions in Cash Flow Analysis
(a) Triangular Distribution (b) Uniform Distribution
L: minimum valueMo: mode (most-likely)H: maximum value
Contemporary Engineering
Economics, 4th edition, © 2007
Discrete Distribution -Probability Distributions for Unit Demand (X) and Unit Price (Y) for BMC’s Project
Product Demand (X) Unit Sale Price (Y)
Units (x) P(X = x) Unit price (y) P(Y = y)
1,600 0.20 $48 0.30
2,000 0.60 50 0.50
2,400 0.20 53 0.20
Contemporary Engineering
Economics, 4th edition, © 2007
Unit Demand
(x)
Probability
P(X = x)
1,600 0.2
2,000 0.6
2,400 0.2
F x P X x x
x
x
( ) ( ) . , ,
. , ,
. , ,
0 2 1 600
0 8 2 000
10 2 400
Cumulative Probability Distribution for X
Contemporary Engineering
Economics, 4th edition, © 2007
Probability and Cumulative Probability Distributions for Random Variable X
Contemporary Engineering
Economics, 4th edition, © 2007
Probability and Cumulative Probability Distributions for Random Variable Y
Contemporary Engineering
Economics, 4th edition, © 2007
E X p xj jj
j
[ ] ( )
1
(discrete case)
(continuous case) xf(x)dx
Measure of Expectation
Contemporary Engineering
Economics, 4th edition, © 2007
Expected Return Calculation
Event Return (%)
Probability Weighted
1
2
3
6%
9%
18%
0.40
0.30
0.30
2.4%
2.7%
5.4%
Expected Return (μ) 10.5%
Contemporary Engineering
Economics, 4th edition, © 2007
2 2
1
( ) ( )j
x j jj
Var X x p
x Var X
Var X p x p xj j j j 2 2( )
E X E X2 2( )
Measure of Variation
Contemporary Engineering
Economics, 4th edition, © 2007
Event Deviations Weighted Deviations
1 (6% - 10.5%)2 0.40 (6% - 10.5%)2
2 (9% - 10.5%)2 0.30 (9% - 10.5%)2
3 (18% - 10.5%)2 0.30 (18% - 10.5%)2
( 2) = 25.65
Variance Calculation
σ = 5.06%
Contemporary Engineering
Economics, 4th edition, © 2007
Example 12.5 Calculation of Mean & Variance
Xj Pj Xj(Pj) (Xj-E[X]) (Xj-E[X])2 (Pj)
1,600 0.20 320 (-400)2 32,000
2,000 0.60 1,200 0 0
2,400 0.20 480 (400)2 32,000
E[X] = 2,000 Var[X] = 64,000
252,98
Yj Pj Yj(Pj) [Yj-E[Y]]2 (Yj-E[Y])2 (Pj)
$48 0.30 $14.40 (-2)2 1.20
50 0.50 25.00 (0) 0
53 0.20 10.60 (3)2 1.80
E[Y] = 50.00 Var[Y] = 3.00
Contemporary Engineering
Economics, 4th edition, © 2007
P x y P x P y( , ) ( ) ( )
P x y P( , ) ( , ,$48) 1 600
P x y P y( , $48 ( $48)
( . )( . )
.
1 600
010 0 30
0 03
P x y P X xY y P Y y( , ) ( ) ( )
Joint and Conditional Probabilities
Contemporary Engineering
Economics, 4th edition, © 2007
Assessments of Conditional and Joint Probabilities
Unit Price Y
Marginal
Probability
Conditional
Unit Sales X
Conditional
Probability
Joint
Probability
1,600 0.10 0.03
$48 0.30 2,000 0.40 0.12
2,400 0.50 0.15
1,600 0.10 0.05
50 0.50 2,000 0.64 0.32
2,400 0.26 0.13
1,600 0.50 0.10
53 0.20 2,000 0.40 0.08
2,400 0.10 0.02
Contemporary Engineering
Economics, 4th edition, © 2007
Xj
1,600 P(1,600, $48) + P(1,600, $50) + P(1,600, $53) = 0.18
2,000 P(2,000, $48) + P(2,000, $50) + P(2,000, $53) = 0.52
2,400 P(2,400, $48) + P(2,400, $50) + P(2,400, $53) = 0.30
P x P x yy
( ) ( , )
Marginal Distribution for X
Contemporary Engineering
Economics, 4th edition, © 2007
Covariance and Coefficient of Correlation
( , )
( [ ])( [ ])
( ) ( ) ( )
( , )
xy
xy x y
xyx y
Cov X Y
E X E X Y E Y
E XY E X E Y
Cov X Y
Contemporary Engineering
Economics, 4th edition, © 2007
Calculating the Correlation Coefficient between X and Y