Lecture 9: Modeling and Simulation

Modeling and Simulation Statistical Models in Simulation LECTURE 8 BY D R. S A M A H A. Z. H A SS A N Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model might well describe the variations. An appropriate model can be de...

Modeling and Simulation Statistical Models in Simulation LECTURE 8 BY D R. S A M A H A. Z. H A SS A N Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model might well describe the variations. An appropriate model can be developed by sampling the phenomenon of interest: ◦ Select a known distribution through educated guesses ◦ Make estimate of the parameter(s) ◦ Test for goodness of fit  Review of Terminology and Concepts We will review the following concepts:  Discrete random variables  Continuous random variables  Cumulative distribution function  Expectation  Useful Statistical models  Discrete Distributions ◦ Bernoulli trials and Bernoulli distribution ◦ Binomial distribution ◦ Geometric and negative binomial distribution ◦ Poisson distribution  Continuous Distributions (1) Discrete Random Variables X is a discrete random variable if the number of possible values of X is finite, or countably infinite. Example: Consider jobs arriving at a job shop. ◦Let X be the number of jobs arriving each week at a job shop. ◦Rx = possible values of X (range space of X) = {0,1,2, …} ◦p(xi) = probability the random variable is xi = P(X = xi) (1) Discrete Random Variables p(xi), i = 1,2, … must satisfy: Example 1  Consider the experiment of tossing a single die. Define x as the number of spots on the up face of the die after a toss, then  Domain(X) = Rx = {1, 2, 3, 4, 5, 6}  Assume the prob. that a given face lands up is proportional to the number of spots showing. The discrete probability distribution for this random experiment is given by: Example 2 (2) Continuous Random Variables If the range Rx of the random variable X is an interval or a collection of intervals, X is called a continuous random variable. For a continuous random variable X, the probability that X lies in the interval [a, b] is given by: The unction f(X) is called the probability density function (pdf) of the random variable X. the pdf satisfies the following conditions: (2) Continuous Random Variables As a result of Equation (1), for any specified value x 0, (2) Continuous Random Variables (3) Cumulative Distribution Function The cumulative distribution function (cdf), denoted by F(x), measures the probability that the random variable X assumes a value less than or equal to x, that is, F(x) = p(X ≤ x). Example The die-tossing experiment, described in Example 2, has a cdf given as follows: Where [a, b] = {a ≤ x ≤ b}. The cdf for this example is shown graphically in Figure 4. If X is a discrete random variable with possible value x 1, x2, …. Where x1