1. Contemporary Engineering Economics, 4 th edition, © 2007 Probabilistic Cash Flow Analysis Lecture No. 47 Chapter 12 Contemporary Engineering Economics Copyright, © 2006

2. Contemporary Engineering Economics, 4 th 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

3. Contemporary Engineering Economics, 4 th edition, © 2007 Types of Probability Distribution Continuous Probability Distribution  Triangular distribution  Uniform distribution  Normal distribution Discrete Probability Distribution

4. Contemporary Engineering Economics, 4 th edition, © 2007 (for a discrete random variable) (for a continuous random variable )  f(x)dx Cumulative Probability Distribution

5. Contemporary Engineering Economics, 4 th edition, © 2007 Useful Continuous Probability Distributions in Cash Flow Analysis (a) Triangular Distribution (b) Uniform Distribution L: minimum value M o : mode (most-likely) H: maximum value

6. Contemporary Engineering Economics, 4 th 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) 1,6000.20$480.30 2,0000.60500.50 2,4000.20530.20

7. Contemporary Engineering Economics, 4 th edition, © 2007 Unit Demand (x) Probability P(X = x) 1,6000.2 2,0000.6 2,4000.2 Cumulative Probability Distribution for X

8. Contemporary Engineering Economics, 4 th edition, © 2007 Probability and Cumulative Probability Distributions for Random Variable X

9. Contemporary Engineering Economics, 4 th edition, © 2007 Probability and Cumulative Probability Distributions for Random Variable Y

10. Contemporary Engineering Economics, 4 th edition, © 2007 (discrete case) (continuous case)  xf(x)dx Measure of Expectation

11. Contemporary Engineering Economics, 4 th edition, © 2007 Expected Return Calculation EventReturn (%) ProbabilityWeighted 123123 6% 9% 18% 0.40 0.30 2.4% 2.7% 5.4% Expected Return (μ)10.5%

12. Contemporary Engineering Economics, 4 th edition, © 2007 Measure of Variation

13. Contemporary Engineering Economics, 4 th edition, © 2007 Variance Calculation σ = 5.06%

14. Contemporary Engineering Economics, 4 th edition, © 2007 Example 12.5 Calculation of Mean & Variance XjXj PjPj Xj(Pj)Xj(Pj)(X j -E[X])(X j -E[X]) 2 (P j ) 1,6000.20320(-400) 2 32,000 2,0000.601,20000 2,4000.20480(400) 2 32,000 E[X] = 2,000Var[X] = 64,000  252,98 YjYj PjPj Yj(Pj)Yj(Pj)[Y j -E[Y]] 2 (Y j -E[Y]) 2 (P j ) $480.30$14.40(-2) 2 1.20 500.5025.00(0)0 530.2010.60(3) 2 1.80 E[Y] = 50.00Var[Y] = 3.00 

15. Contemporary Engineering Economics, 4 th edition, © 2007 Joint and Conditional Probabilities

16. Contemporary Engineering Economics, 4 th edition, © 2007 Assessments of Conditional and Joint Probabilities Unit Price Y Marginal Probability Conditional Unit Sales X Conditional Probability Joint Probability 1,6000.100.03 $480.302,0000.400.12 2,4000.500.15 1,6000.100.05 500.502,0000.640.32 2,4000.260.13 1,6000.500.10 530.202,0000.400.08 2,4000.100.02

17. Contemporary Engineering Economics, 4 th edition, © 2007 XjXj 1,600P(1,600, $48) + P(1,600, $50) + P(1,600, $53) = 0.18 2,000P(2,000, $48) + P(2,000, $50) + P(2,000, $53) = 0.52 2,400P(2,400, $48) + P(2,400, $50) + P(2,400, $53) = 0.30 Marginal Distribution for X

18. Contemporary Engineering Economics, 4 th edition, © 2007 Covariance and Coefficient of Correlation

19. Contemporary Engineering Economics, 4 th edition, © 2007 Calculating the Correlation Coefficient between X and Y

20. Contemporary Engineering Economics, 4 th edition, © 2007 Meaning of Coefficient of Correlation Case 1: Case 2: Case 3: