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by: Giovani Ullrich PhD


Giovani Ullrich PhD
GPA 3.5


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Class Notes
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This 21 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 322 at Iowa State University taught by Staff in Fall. Since its upload, it has received 49 views. For similar materials see /class/214411/stat-322-iowa-state-university in Statistics at Iowa State University.




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Date Created: 09/26/15
EXPECTATION MEAN VARIANCE OUTLINE 0 Review of Concepts of Random Variables and PMF 0 Functions of Random Variables o Expectation Mean Variance Reading Bertsekas amp Tsitsiklis 23 24 EESTAT 322 7 FUNCTIONS OF RANDOM VARIABLES 0 Suppose we have a RV X defined on an experiment Le a function Xw Furthermore suppose that we have a function of the random variable Y gX such as X2 608X 63X Then these are also random variables defined on the original experiment by Yw gXw Le a function of a function isjust another function 0 Since Y isjust a random variable it must have a PMF pyy which we can find from the original probability model 0 There is a more direct and usually simpler way find PMF for Y gX from the PMF of X EESTAT 322 7 2 FUNCTIONS OF RANDOM VARIABLES Sample set of ossible By set x gxy 3 pitn3fi 13 es IPMF my PAy W in Byl laxy 130 1 WW I 3 2 PM w9Xwy Z Z PM Z PM mgmywXwm 1vgvy EESTAT 322 7 EXAMPLE 0 X maximum of two rolls of a foursided die 0 YgX as 9196 1 23 0 otherwise 0 Find the PMF of Y PYCU Z PXW mgy 7 5 3 1 PY WEngXW 16 16 4 py01 py1i EESTAT 322 7 EXPECTATION Expectation The expected value expectation or mean of a random variable X with PMF pX is defined as EX Zach center of gravity c mean EX EESTAT 322 7 VARIANCE MOMENTS STANDARD DEVIATION 0 2nd Moment EX2 nth Moment EXquot o Variance varX 1 E X EiXi2 0 Always nonnegative 0 Standard deviation 0X varX EESTAT 322 7 EXPECTED VALUE RULE FOR FUNCTIONS OF RANDOM VARIABLES 0 Let Y gX be a function of the RV X Em E9X Zgpx E9X Em Ewany 22 2 MW 9 gy Z Z prxZ Z gxpxxZgxpxltx y gy y mlgy EESTAT 322 7 EXPECTED VALUE RULE CONT D o Linearity of the expectation Ei91X 92Xi Ei91Xi Ei92Xi o Variance varX EX EX2 EX2pXx o Moments EXquot Em 90an90 0 Variance in terms of moments varX EX2 EX2 0 Mean and variance of a linear function of a RV 0 Let Y CLX i b o EY aEX b varY agvarX EESTAT 322 7 RANDOM VARIABLES OUTLINE o Other PDFs 7 Exponential and Erlang 7 Delta 0 Problem Solving Examples Reading G R Cooper amp C D McGillem 26 27 EESTAT 322 7 EXPONENTIAL AND RELATED DISTRIBUTIONS The PDF is given by 1 7LT i e T w 2 0 lew T l 0 otherwise EX f and a 2 EX fooo fwdw fooo 67dew By integrating by parts wdy wyl i fydw we get EX 70 m m 1 0 iweT lgo 6 0 IllH Idw EESTAT 322 7 EXPONENTIAL DISTRIBUTION CONT o Exponential distribution is memoryless FX g t 1 i 67 Let PrX gt31i FXt e t then PrX gt t le gt s Pr rgtltt gtsgt e ltt5gtes 67 PrX gt t In other words a device to survive another If time is independent of how long it has been used as if it forgets it has been used for s time Vice versa if PrX gt 25 le gt s PrX gt 25 holds for all s and 25gt 0 then X must have an exponential distribution EESTAT 322 7 3 EXPONENTIAL DISTRIBUTION CONT Example The waiting time of a customer has an exponential distribution with a mean of 5 minutes Then what is the probability that he will wait more than 10 minutes Solution 5 5 PrX gt 10 10 e 2 01353 If he already spent five minutes there the chance that he needs to wait another 10 minutes is given by PrX gt 15lX gt 5 615765 PrX gt 10 01353 EESTAT 322 7 4 ERLANG DISTRIBUTION Exponential distribution arrival interval between the first and second customers fig a Erlang arrival interval between the first and the fourth customers fig b gtllt events 17 X I C x n s 39 39 39 39 i I L I p Z t a b EESTAT 322 7 5 ERLANG DISTRIBUTION CONT The PDF with parameter k is given by Tk71eix fXwk 95301 2a3aw x lt 0 ln gamma distribution the PDF is we expe rww When we let 1 and 04 k gamma a Erlang EESTAT 322 7 6 DELTA DISTRIBUTION 39 PDF 1635 171535 i 351 1725l 352 where p1 p2 1 and 6 is the Kronecker delta function 0 Example Two possible outcomes of a coin experiment 351 0 for H and 352 1 for T and p1 p2 05 ElX Y ffooolp16 1 P25 362d36 P1361 P2362 EX2 fix wzlpl w 1 p26 2dw plat 192353 72 03 EX2 X p1p21 gm EESTAT 322 7 DELTA DISTRIBUTION CONT 0 Extension Multiple outcomes ag Elfin6amp5 i where 271191 1 Y 2211 Pill W 2211 03 ZL1 mamm 92 Example Ex 2 73 Three coins are tossed Let H number of heads Find 0 EESTAT 322 7 DELTA DISTRIBUTION CONT Solution We have four possible outcomes HO 0 with probability p0 3 13 H1 1 with probability p1 33 H2 2 with probability p2 33 H3 3 with probability p3 13 The PDF l5 NH Ziopi H HiI EHpo0p11p22p3312315 EH2po0p112p222p3322433 0 EH2 a EH2 3 7152 075 EESTAT 322 7 9 EXAMPLES FOR DENSITY FUNCTIONS Example Problem 2 45 textbook RV X has a pdf of the form fXw m2 0lt 2 am 2 lt m g 3 Find a the value of a b c P1quot2 lt X g 3 Solution 3 We need f0 fX3cdac 1 so that L awzdw f aw MB3 8 a22l a83 92 421 a a 631 b EX fOZ wawzdw wawdw 34 9 7 83 2 c Pr2 lt X g 3 ff fXd 23wdw 1531 EESTAT 322 7 10 EXAMPLES FOR DENSITY FUNCTIONS CONT Example 2 53 Gaussian RV X has a probability of 05 of having value less than or equal to 1 Further PrX gt 50 00228 Find a EX b 0 c PrX g 3 Solution a PrX 1 05 So EX1 b PrX gt 5 17 ltIgt5 1039I Q5 1039I 00228 Using the inverse of Q function we get 40 2 so a 4 c PrX g 3 ltIgt3 10I ltIgt1 08413 Alternatively PrX g 3 17PrX gt 3 17Q3710m17Q1 03413 EESTAT 322 7 11 Example 2 55 False alarm the observation is over a threshold with noise input only Detection the observation is over a threshold when the signal is present Gaussian noise input n with zero mean and variance of 1 Threshold 5 Find a PrFaIse alarm b A signal 3 8 is present find Prdetection Solution a 0 021 PrFaIse alarm Prn gt 5 Q5 i 01 Q5 287 10 7 b Signal 3 8 a 0 Combined input Es 11 Es 8 02 1 0Prggl etection Prs n gt 5 1 ltIgt5 i 80 1 73 Alternatively Prs n gt 5 Q5 i 80 Qi3 1 Q3 09987 EESTAT 322 7 12 Example 2 63 A current I with a Rayleigh PDF passes through a resistor with R 27TSZ EI 2 A Power dissipation W R12 a Find the mean of power dissipated b Find P1quotW g 12 c P1quotW gt 72 Solution 3 EI2WUHU2 27 REUZ R202 2W2 4 27T 32 b PrW g 12 MRI2 g 12 13112 lt 191 PM lt 1332 FI1382 17 exp7139 222 03127 c PrW gt 72 PrI gt 3335 1 i F13385 01054 EESTAT 322 7 13 EXAMPLES FOR DENSITY FUNCTIONS CONT Example 2 71 RV 6 is uniformly distributed in 0 2W Another RV X is given by X 0056 a Find PDF of X b Find EX c Find 0 d Find PrX gt 05 Solution a elmd6 781116 7V1 7 cos2 7V1 7 2 EESTAT 322 7 14 EXAMPLES FOR DENSITY FUNCTIONS CONT ac c056 has two roots solutions in 0 2W given by 61 0013 1 w E 07T and 62 27T cos w 6 7T 2W EESTAT 322 7 15 EXAMPLES FOR DENSITY FUNCTIONS CONT At both 61 and 62 Id1 d6I V17 352 2 f6 1 fXWWW 71931 0 elsewhere b EX Ecos6 fOZWcos6d6 0 C EX2 W EICOS2 0 1 02 c0526d6 a jifz 72 EESTAT 322 7 16 EXAMPLES FOR DENSITY FUNCTIONS CONT d PrX gt 05 f0 fXwdw fol dx 13 Procedure Let ac cos6 for 6 E 07T2 So clan sin 6d6 V117 5519 w 14 6 00841 0 w 05 a 6 cos 105 7T3 1 0 Aka dw 7 sin 6d6 3 7Tsin6 1 7r31 id613 0 7T EESTAT 322 7 17 SUMMARY AND QUESTIONS Elementary Set Theory Axiomatic Approach 7 Set operations union intersection complement subtraction probability space independence mutually exclusive distributive law De Morgan law 7 Total probability theorem PrB 2171 PrBlA PrA Bayes rule PrAlB PrlBL grlAD PNBIADPNAD 7 221 PNBIAi PNAD39 Conditional Probability lndependence Application of Bernoulli Trials Examples error correction coding system reliability random le false positive puzzles circuit switches EESTAT 322 13 SUMMARY AND QUESTIONS CONT 0 Random Variable Distribution Functions PDF Mean Values and Moments 7 Gaussian mean and variance related PDFs X N NM02 relation between w and FltXgt Wt Qw1 62w and Qw1 mac Rayleigh Exponential RVs Other PDFs uniform quantization error and MSE EESTAT 322 13 SUMMARY AND QUESTIONS CONT 7 One PDF to another PDF From fXgc 7 fyy when 3 fyy Wfg 1y Example 1W9H fwy w sing 7 Conditional distributionsPDFsmean ying M X g m Example X NX02 M X g X EESTAT 322 13 3 SUMMARY AND QUESTIONS CONT 0 Two Random Variables 7 Joint Probability distribution joint PDF Conditional Probability 7 Bayes Rule Estimation Based on Observation 7 Statistical Independence Correlation Between Random Variables 7 Variance of Sum X Y PDF of Z X Y PDF of Z gX Y gX Y XY Jacobian factor J 0 Characteristic Function CHF 7 Obtain CHF from PDF Obtain CHF for Z X1 X2 X3 7 Obtain moments from CHF EXiYk EESTAT 322 13 4 SUMMARY AND QUESTIONS CONT Example 1 A RV X has a PDF of the form fXw expikw gt0 i 0 elsewhere and Y 7 X239 1 Find the PDF of Y fyy 2 Find the probability that Y Z 3 EESTAT 322 13 5 SUMMARY AND QUESTIONS CONT 467495 w 2 0 Example 2 A RV X has a PDF ag 0 elsewhere a Find the characteristic function CHF of X b Find the first and second moments of X using the CHF EESTAT 322 13 6 SUMMARY AND QUESTIONS CONT Example 3 Two RVs X and Y are independent X have a PDF of fi 2170 1 X 36 7 0 elsewhere Y is uniformly distributed between 1 and 1 3 Find the PDF of the RV Z X Y b Find the probability that 0 lt Z g 1 EESTAT 322 13 7


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