# MSU CMPS 4223 - Lecture Notes (28 pages)

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## Lecture Notes

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- Pages:
- 28
- School:
- Midwestern State University
- Course:
- Cmps 4223 - Intro To Simulation

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Statistical Statistical Distributions Distributions Uniform Distribution A R V is uniformly distributed on the interval a b if it probability function Fully defined by a b P x 1 b a for a x b 0 otherwise Uniform Distribution Probability Function 1 1 9 1 10 Probability that x is between 2 and 7 5 Probability that x 8 1 1 9 1 10 Uniform Distribution The cumulative distribution of a uniform RV is F x 0 for x a x a b a for a x b 1 otherwise Uniform Distribution Cumulative Function 1 1 10 Uniform Distribution Discrete vs Continuous Discrete RV Number showing on a die Continuous RV Time of arrival When programming make it discrete to some number of decimal places Uniform Distribution Mean a b 2 Variance b a 2 12 P x X y F y F x y a x a y x a a y x b a b a b a b a Uniform Example A bus arrives at a bus stop every 20 minutes starting at 6 40 until 8 40 A passenger does not know the schedule but randomly arrives between 7 00 and 7 30 every morning What is the probability the passenger waits more than 5 minutes Uniform Solution X RV Uniform 0 30 i e 7 00 7 30 Bus 7 00 7 20 7 40 Yellow Box 5 minute wait 1 1 30 A 5 40 B 10 15 C 20 25 30 P x 5 A C 1 B 5 6 Arithmetic Mean Given a set of measurements y1 y2 y3 yn Mean y1 y2 yn n Variance Variance of a set of measurements y1 y2 y3 yn is the average of the deviations of the measurements about their mean m i 1 n V 2 1 n yi m 2 Variance Example Yi 12 10 9 8 14 7 15 6 14 10 m 10 5 V 2 1 10 12 10 5 2 10 10 5 2 1 10 1 52 52 1 52 1 10 88 5 8 85 Standard Deviation 2 975 Normal Distribution Has 2 parameters Mean Variance 2 Also Standard deviation Normal Dist 3413 1359 0215 0013 3 2 1 1 2 0 Mean n 3 Normal Distribution Standard Normal Distribution has Mean 0 StdDev 1 Convert non standard to standard to use the tables Z value of StdDev from the mean Z is value used for reading table Z x m Normal Example The scores on a college entrance exam are normally distributed with a mean of 75 and a standard deviation of 10 What of scores fall between 70

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