1. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 CSE474: Simulation and Modeling Chapter 4 Review of Basic Probability and Statistics Mushfiqur Rouf nasarouf@gmail.com

2. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Why How to model a probabilistic system Validate a simulation model Choose an input probability distributions Generate random samples from these distributions Perform statistical Analyses of the simulation output data

3. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Experiment –A process whose outcome is not known with certainty –Throwing a die Sample Space, S –Set of all outcomes –{1, 2, 3, 4, 5, 6} Sample Point –Each outcome in a sample space

4. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Random Variable A function that assigns a real number to each point in sample space S. –If X = “number of heads” in an experiment of rolling a pair of dice. –Then X assigns 5 to {4, 1}, {3, 2}, {2, 3}, {1, 4} Discrete: if it can take countable number of different values Continuous: it can take any value

5. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Probability Distribution Function Or, cumulative distribution function F(x) of random Variable X –X: random variable name –x: value taken F(x) = P(X<=x) for –∞ < x < ∞ Properties –0 <= F(x) <= 1 –F(x) is nondecreasing – and

6. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Probability Distribution Function X can take values –x 1, x 2, …, x n, Probability mass function “probability that x equals to x i ” p(x i ) = P(X = x i )

7. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Probability Distribution Function 1/6 1/3 1/2 0 1 2 3 4 x p(x) 0 1 2 3 4 x 1 1/6 1/3 1/2 F(x) 1

8. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Continuous Random Variable X is a continuous random variable if probability density function f(x) is nonnegative

9. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Continuous Random Variable f(x) is not the probability that X=x X is more likely to fall in an interval I

10. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Continuous Random Variable Distribution function F(x) –Area under the curve f(x) f(x)f(x) x F(x) = P(X  [- , x]) f(x)f(x) b P(X  [a, b]) = F(b) – F(a) ax

11. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Uniform Random Variable 0 <= x <= 1 otherwise 1x0 1 f(x)f(x) 1x0 1 F(x)F(x) U[0,1]

12. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Joint Probability Mass Function p(x, y) = P(X = x, Y = y) X and Y are independent if p(x, y) = p x (x) p y (y) Calculate if X and Y are independent For x = 1, 2 and y = 2, 3, 4 otherwise

13. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Jointly Continuous Joint probability density function X and Y are independent if

14. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Mean or Expected Value E(cX)=cE(X)

15. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Median Smallest value of x such that F Xi (x) >= 0.5 median F(median) = 0.5 area = 0.5

16. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Variance μμ σ 2 large σ 2 small Calculate Mean and Variance of U[0,1]

17. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Standard Deviation σ i = √(σ i 2 ) Useful with Normal distribution

18. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Covariance Dependence between two random variables C ij = 0 means X i and X j are uncorrelated C ij > 0 means X i and X j are positively correlated C ij < 0 means X i and X j are negatively correlated

19. CSE 474 Simulation Modeling | MUSHFIQUR ROUF nasarouf@gmail.com http://groups.google.com/group/cse474spring07/ http://faculty.bu.ac.bd/~rouf/cse474 Correlation Covariance is not dimensionless, –makes interpretation troublesome