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Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events >0 Memoryless property IME 312 Exponential and Poisson relationship Unit Matching between x and ! f(x) F(x)

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Exponential Dist. Poisson Dist. IME 312

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X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8 0 1:00 2:00 3:00 Y1=3 Y2=4 Y3=5 Relation between Exponential distribution ↔ Poisson distribution X i : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 Y i : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4) IME 301 and 312

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Continuous Uniform Distribution f(x) IME 312 F(x) a a b b

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Gamma Distribution K = shape parameter >0 = scale parameter >0 IME 312 For Gamma Function, you can use: and if K is integer (k’) then:

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Application of Gamma Distribution K = shape parameter >0 = number of Y i added = scale parameter >0 = rate if X ~ Expo ( ) and Y = X 1 + X 2 + ……… + X k then Y ~ Gamma (k, ) i.e.: Y is the time taken for K events to occur and X is the time between two consecutive events to occur IME 312

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X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8 0 1:00 2:00 3:00 Relation between Exponential distribution ↔ Gamma distribution X i : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 Y i : Continuous random variable, time taken for 3 customers to arrive, has Gamma distribution with shape parameter k = 3 and scale=4 IME 312 Y 1 =1 Y 2 =7/8 Y 4 =3/4 Y 3 =1/2

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Weibull Distribution a = shape parameter >0 = scale parameter >0 for IME 312

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Normal Distribution IME 312 Standard Normal Use the table in the Appendix

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Normal Approximation to the Binomial Use Normal for Binominal if n is large X~Binomial (n, p) IME 312 Refer to page 262

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Central Limit Theorem : random sample from a population with and : sample mean Then has standard normal distribution N(0, 1) as commonly IME 312

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What does Central Limit Theorem mean? Consider any distribution (uniform, exponential, normal, or …). Assume that the distribution has a mean of and a standard deviation of. Pick up a sample of size “n” from this distribution. Assume the values of variables are: Calculate the mean of this sample. Repeat this process and find many sample means. Then our sample means will have a normal distribution with a mean of and a standard deviation of. IME 312

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= degrees of freedom = probability Distribution Definition Notation Chi-Square t dist. F dist. Where: IME 312

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Some standard univariate probability distributions Characteristic function, moment generating function, cumulant generating functions Discrete distribution.

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