Stat400Lec11(Ch3.5,3.6)_ans

# Stat400Lec11(Ch3.5,3.6)_ans - stot 400 lrecture I I Silins...

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stot 400 lrecture I I Silins 2012 Review (3.4,3.7) o Uniform distribution, its mean, variance, m.g.f. r Exponential distribution and its mean, variance. o Memoryless properties of exponential distribution. o Cauchy distribution, GOT A MOMENT? Today's Lecture (3.5,3.6) r Gamma distribution and its connection to Poisson Process o Mean, variance and m.g.f. of Gamma distribution. r x2 distribution. o Normal (Gaussian) distribution Stat 400 tccture ll Spriry 2012 Gamma function f(t):f-yt-te-Ydy, ,>0 Properties of t(t): r r(t) : (, - 1)r(, - 1),, > 1. o f(1) : tf; e-udy : 1. o When t: n, a positive integer, r(") : r(n, - 1)(n, - 1) : ... : (n - 1)! Gamma distribution Definition: The random variable X has a gamma distri- bution if its p.d.f. is defined by 1 f (r): ffir"-re-*/o, o S r < oo Write Gamma(o, 0), where 0 : ll),, a (not neces- sarily an integer). 9t- Prrt [,Lt> stat 400 l4ture ll Volcano eruption (again) Let I{r be the number of volcano eruptions to have oc- curred by time t, starting from now. Suppose that the volcano eruption forms a -Poisson process with rate ,\. Then Nr - Let X be the waiting time until the 4-th volcano eruption occurs and find the distribution of X. r^_,a/--\a. :-_> F,b\ :? ({, (Ml,-Y ( N^az, +\ = l--'? (M^ s r ) - \st =rr/{(r+}n+ry'.4) fr,x> = tltg : /n[-,r--,t'x* *:- y' + .\ +r? + y'] : PJf ,U ln general, if X is the waitiig tirnu ,itit the a-th volcano eruption, then Fx(r):1-'i'@)F,,,o

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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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Stat400Lec11(Ch3.5,3.6)_ans - stot 400 lrecture I I Silins...

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