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# PROBABILITY I STAT 394

UW

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This 2 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 394 at University of Washington taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/192514/stat-394-university-of-washington in Statistics at University of Washington.

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Date Created: 09/09/15

Lecture 28 More functions of continuous random variables 281 The log of a Uniform random variable Suppose X is Uniform 01 and let Y 710g X Note 0 lt X lt 1 so 0 lt Y lt oo Fyy PY y P10gX y 1310ng 7y 7 P X 7 1 1d 1 7 gtex 7 7 78X 7 7 p 24 AXPH p 24 WW Pity 7 expky 0ltyltltgt0 That is Y is an exponential random variable with parameter 1 282 Maxima and minima of independent random variables a Suppose X1 X2 are independent and have cdf F1m F2m Let Y maxlgign Xi 130 S y Pall the X are g y TL PXy 7 HEW i1 H 13 H because the X are independent b Now with the same Xi let W mlnlgign 17 PY gt y Pall the X are gt y V L V L HPOQ gt y H1Fz y i1 13971 283 The Max of uniform random variables Suppose X are independent and all are Uniform 01 rvs Let Y maxlgign Xi Then z and Fyz PY z z Note that as n 7 00 m 7 O Y piles up the probability closer and closer to 1 So instead let us consider V n17 Y PV U Pl 7 Y vn PY 217 vn 17Fy1 71171 17 1 7 3 7gt17exp7v 71 That is fvv exp7v V is exponential with parameter 1 284 The Minimum of exponential random variables Suppose Xa X5 and X0 are independent exponential rvs with rates Au b and Ac Let Y minXa X5 X0 For example X are waiting times for busses a is No 48 b is No72 c is No 373 17 PY gt y Pall the X are gt y PXa gt yPXb gt yPXc gt 00 00 Now PXa gt y may Aaexp7xazdz 7 exp7az exp7ay y 2 SO 1 Fifty expayexpbyexpcy expa b My My FHy A A1 Acexp7Aa A1 My That is Y is an exponential random variable with rate parameter a b Ac This makes sense if I can take any of the three busses collectively they arrive at the sum of the rates and my expected waiting time is 1rate Lectures 293031 REVIEW Review of conditional probability Midterm 2 question 1 Review of Bernoulli process Midterm 27 question 3 Another Bernoulli process example Homework 87 Ch 37 No79 Other examples 7 ask Crib sheet for nal7 essentially as for midterm 2 Plus a Normal probability table will be provided

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