Contemporary Engineering Economics, 4 th edition, © 2007 Probabilistic Cash Flow Analysis Lecture No. 47 Chapter 12 Contemporary Engineering Economics.

Contemporary Engineering

Economics, 4th edition, © 2007

Probabilistic Cash Flow Analysis

Lecture No. 47Chapter 12Contemporary Engineering EconomicsCopyright, © 2006



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



Types of Probability Distribution Continuous Probability Distribution

Triangular distribution Uniform distribution Normal distribution

Discrete Probability Distribution



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



Useful Continuous Probability Distributions in Cash Flow Analysis

(a) Triangular Distribution (b) Uniform Distribution

L: minimum valueMo: mode (most-likely)H: maximum value



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



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



Probability and Cumulative Probability Distributions for Random Variable X



Probability and Cumulative Probability Distributions for Random Variable Y



E X p xj jj

j

[ ] ( )

1

(discrete case)

(continuous case) xf(x)dx

Measure of Expectation



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%



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



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%



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



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



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



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



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



Calculating the Correlation Coefficient between X and Y



Meaning of Coefficient of Correlation Case 1:

Case 2:

Case 3:

Contemporary Engineering Economics, 4 th edition, © 2007 Probabilistic Cash Flow Analysis Lecture No. 47 Chapter 12 Contemporary Engineering Economics.

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assessment of probability

probability concepts

random variable y slide

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maximum value slide

unit demand x probability