IE 303 Discrete-Event Simulation

Transcription

1 IE 303 Discrete-Event Simulation 1 L E C T U R E 5 : P R O B A B I L I T Y R E V I E W

2 Review of the Last Lecture Random Variables Probability Density (Mass) Functions Cumulative Density Function Discrete Distributions Bernoulli Distribution Binomial Distribution Geometric Distribution Negative Binomial Distr. Poisson Distribution 2

3 Outline Continuous Distributions Uniform Distributions Exponential Distributions Gamma and Erlang Distributions Weibull Distribution Normal Distribution Triangular Distribution Lognormal Distribution Truncated Normal Convolution Empirical Distributions Maximum Likelihood Estimation 3

4 Cont. Distr: Uniform

5 Cont. Distr: Exponential

6 Exponential Distribution Exponential Distribution has memoryless property. 6 Conditional probability??? Example: Let X represent the life of a component (a battery, light bulb, computer chip, etc) and assume that X is exponentially distributed. The property states that, given that the component is observed to have survived s hours, the probability that it will survive t more hours is the same as the initial probability that it lives at least t hours. Only the exponential and geometric distributions possess this property.

7 Gamma Distribution

8 Gamma Distribution

9 Erlang Distribution 9 Recall that Poisson distribution counts arrivals with exponential interarrival times. Hence there is a relationship between Erlang and Poisson distributions: Take a time interval S=[0,x]. Define X~ Erlang(k,λ) and Y~Poisson(λx). means that there are at least k i.i.d. exponentials (with parameter λ) exists in S. Hence the number of exponentials in S should be larger-than-equalto k.

10 Weibull Distribution 10 Similar to the gamma distribution, the first two distribution parameters of the Weibull distribution α and β called scale and shape parameters respectively. The third parameter, ν, is called the location parameter. Notice that, as mentioned before, with β = 1 and ν = 0, the Weibull random variable becomes an exponential with rate λ = 1/α.

11 Weibull Distribution

12 Normal Distribution Recall that μ is location, σ is the scale parameter for Normal Distr.

13 Normal Distribution For calculations with Normal distributions, we use standard normal distribution with μ=0, σ=1. 13 Probabilities from normal distribution are calculated as follows:

14 Normal Distribution

15 Triangular Distribution 15

16 Lognormal Distribution

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18 Truncated Normal 18 where a and b constitute the truncation range, and, are parameters of the parent Normal distribution. Idea: Set probability function to zero for values outside the range, scale the rest of the density accordingly. Expectation (μ) and variance (σ)

19 Truncated Normal Distribution (a=0, b= )

20 Convolution So far we covered random distributions. What about we sum two random variables: Suppose random variables X and Y with distributions f(x), g(y) defined on sets Ω x and Ω y. If X and Y are discrete: 20 Recall for independent r.v. A, B If X and Y are continuous:

21 Convolution Example: Suppose X ~ Pois(λ) and Y~Pois(μ). What is the distribution of Z=X+Y? 21 Recall and Example: Show that if W~Exp(λ), then V=W+W ~ Erlang(2,λ). Recall Erlang distribution:

22 Maximum Likelihood Estimation 22 So far we discussed distributions and their summation. But given a set of data, how do we estimate parameter of the distribution?? Example: For his simulation project Harry collects data for the number of people arriving to D/K building of the university between 8-10 am. He assumes that number of arrivals follows Poisson distribution with parameter λ. How to estimate λ? If a sample of random variables, x i, come from the distribution function f(x,θ), then the likelihood function for this sample is as follows: θ value maximizing L(θ) is called the maximum likelihood estimator of θ.

23 . Maximum Likelihood Estimation If Harry collects the following data for 5 days, what is his estimation of λ? #of Arrival Day1 2 Day2 3 Day3 4 Day4 2 Day Can we derive the estimator in general? Average of the sample is MLE estimator for Poisson distribution. Recall if X~Poisson(λ),

24 Empirical Distribution 24

25 Empirical Distribution 25

26 Empirical Distribution 26

27 Excel Commands for Distributions 27

28 Excel Commands for Distributions 28

29 Excel Commands for Distributions 29

30 30 END OF LECTURE 5 Next Lecture Random Number Generation (Chapter 7)

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