Stat400Lec11(Ch3.5,3.6) - Stat 400 Lecture 11 Spring 2012...

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Unformatted text preview: Stat 400 Lecture 11 Spring 2012 Review (3.4,3.7) Uniform distribution, its mean, variance, m.g.f. Exponential distribution and its mean, variance. Cauchy distribution, GOT A MOMENT? Today's Lecture (3.5,3.6) Gamma distribution and its connection to Poisson Process Mean, variance and m.g.f. of Gamma distribution. 2 Memoryless properties of exponential distribution. distribution. Normal (Gaussian) distribution 1 of 9 Stat 400 Lecture 11 Spring 2012 Volcano eruption (again) Let Nt be the number of volcano eruptions to have occurred by time t, starting from now. Suppose that the volcano eruption forms a Poisson process with rate . Then Nt Poisson( . Let X be the waiting time until the 4-th volcano eruption occurs and find the distribution of X. FX (x) = In general, if X is the waiting time until the -th volcano eruption, then FX (x) = 1 and fX (x) = 0 FX (x) 1 X1 ( x)k e k! k=0 = x e x , x>0 x ( 1)! , x>0 2 of 9 Stat 400 Lecture 11 Spring 2012 Gamma function (t) = Properties of (t): (t) = (t (1) = 0 Z 1 t 1 y e dy, 0 y t>0 Need to know the properties 1) (t 1), t > 1. R1 e y dy = 1. 1)(n 1) = = (n 1)! When t = n, a positive integer, (n) = (n Gamma distribution Definition: The random variable X has a gamma distribution if its p.d.f. is defined by 1 f (x) = x 1e x/ , 0 x < 1 () Write Gamma(, ), where = 1/ , > 0 (not necessarily an integer). how long are you going to wait until alpha event... M (t) = 1 , t < 1/ (1 t) = M 0(0) = 00 2 = M (0) [M 0(0)]2 = ( + 1)2 22 = 2 3 of 9 Stat 400 Lecture 11 Spring 2012 Example Telephone calls enter a college switchboard at a mean rate of 1/2 call per minute according to a Poisson process. Let X denote the waiting time until the second call arrives. What is the distribution of X? X~Gamma( a, =Gamma(2,2) What is the average waiting time? average waiting time 2*=4 What is the probability that the waiting time is longer than 3 minutes? PX>3)=1-F(3) = also can use Poisson. P( N3<2)=P(Poisson(3*1/2<2) from the counting process, we can see counts less than 2 4 of 9 Stat 400 Lecture 11 Spring 2012 Chi-square distribution Gamma distribution: 1 f (x) = x 1e x/ , 0 x < 1 () Chi-square distribution: = 2, = r/2, r is a positive integer. know the relationship between Chi-square to Poisson And also the others 1 f (x) = xr/2 1e x/2, 0 x < 1 (r/2)2r/2 r: degree of freedom. Write X Mean: = = (r/2)2 = r Variance: Plot: 2 2 (r). = 2 = (r/2)22 = 2r. 2t) r/2 m.g.f. M (t) = (1 , t < 1/2. 5 of 9 Stat 400 Lecture 11 Spring 2012 6 of 9 Stat 400 Lecture 11 Spring 2012 7 of 9 Stat 400 Lecture 11 Spring 2012 X~N (36000, 5000^2 PX<42000)=P(Z=(x-36000)/5000)<((42000-36000)/5000) P(X>40000)= 8 of 9 Stat 400 Lecture 11 Spring 2012 9 of 9 ...
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This note was uploaded on 03/14/2012 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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