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# PROBABILITY II STAT 395

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This 10 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 395 at University of Washington taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/192510/stat-395-university-of-washington in Statistics at University of Washington.

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Date Created: 09/09/15

Lecture 15 Feb 9 Expectations Variance and Covariance Ross 71 74 151 Expectations of functions and sums i For a discrete random variable EgX EweY z For a continuous random variable EgX f1 gz fXm dm This was proved for the discrete case in 394 Ross R145 If X and Y have joint pmf pxyy or pdf fxyy then EgXY Em ZygXY pxyy or EgX Y f1 fygX Y fxymy dz dy respectively This would be proved exact same way ll Recall also from 394 EltglltXgt gzltXgtgt 7 MW 9290fX90 dz 7 EltglltXgtgt E92X Now if X and Y have joint pmf pxy y or pdf fxyy then EltglltXgt 920 7 lt91ltXgt 92ltYgtgtfxyltzygtdzdy 7 xgtXyltzygtdydm gzltYgtXyltzygtdzdy X 91 7 gm gtfxltzgtdz 92Yfyydy 7 Elt91ltXgtgt Elt92ltYgtgt 152 Expectation of a product of functions of independent rvs If X and Y are independent random variables Elt91ltXgt92ltYgtgt 7 lt91ltXgt92ltYgtgtfxymwzdy 7 lt91ltXgt92ltYgtgtfxltzgtfyltygtdzdy 7 glltXgtfXltzgtdzmomma 7 EltglltXgtgtEltgzltYgtgt The proof for discrete random variables is similar 153 Variance Covariance and correlation 1 Recall if EX 1 varltXgt 7 EltltX 7 m2 7 M2 7 M 2 7 EltX2gt 7 2MEX M2 7 EltX2gt 7 MW ii If EX u and EY 1 de ne covX Y E EX 7 uY 7 Then covX Y EXY 7 uY 7 VX W EXY 7 uEY 7 VEX W EXY 7 EXEY Note varX covXX and covX7Y 7covX Y iii We see from 152 that if X and Y are independent then covX Y 0 iv The converse is NOT true ie covX Y 0 does not imply XY independent Example X and Y uniform on a circledisc X cosU Y sinU where U N U0 27139 V De ne the correlation coe cient p by pXY covXY varXVarY Note from iii if X and Y are independent pXY 0 As in iv in general the converse is NOT true Also note pXX 1 pX7X 71 We shall show below that 71 g p g 1 always Lecture 16 Feb 11 Variances and covariances of sums of random variables Ross 74 161 Variance and covariance of a sum i Let X have mean 1 i 17n7 and have mean lj j 17m So EZXi ZELM and E63 Y1 Elquot Vi cov 7 7 fig gt mi i1 ZcovXlYj M N H Em Xi MHYJ39 VD En Em EXi MHYJ39 VD 71 iivarltE Xi cov Xi E Xj E E covXlXj E varXi 2 E E ovXlXj i1 39 39 39 i1 H x H H x H ilt7 iii If X and Xj are independent for all pairs Xi7 Xj7 then covXlXj 0 so 71 71 var Xi ZvarXi 391 391 162 The correlation inequality 1 1 Let X have variance 0 and Y have variance 032 Y var X var Y cov XY gt i2 21ipX7y 0X TY axay Hence 0 g 17 pXY so p 10 1 pXY so p 2 71 ie 71 g p g 1 163 Mean and variance of a sample mean Let X17 Xn be independent and identically distributed iid each with mean Ia and variance 02 The sample mean is de ned as Y n 1 21 Xi Then Em 7 Berlin 7 vim 7 WW 7 i i1 i1 V L V L var7 varn 1 n72 ZvarXi n02n2 7271 i1 i391 We can estimate Ia by Y and the variance of this estimator is azn but now we need to estimate 02 164 Mean of a sample variance Let X17 Xn be iid each with mean Ia and variance 02 Note EZiXi 7 p2 n02 but we usually do not know Ia The sample variance is de ned as 2 SAX 7 7 1 Then V L XFYV ZltXFMM72 i1 n 7 1SZ E H ltXi 7i 7 X 7igtlt77igt X7 to H H Xi7 H 717 7 W M Xi NZ 27 M H i1 i71 7 ice 7 NZ 7 277 m7 7 M 7477 NZ 7 ice 7 NZ 7 7477 NZ i1 i1 E092 n i 1 1 EXi NZ nE7 MW i1 n 7 1 1nvarXi 7 nvar7 n 71 1na2 7 n02n 02 Lecture 17 Feb 13 Moment generating functions Ross 77 171 De nition and basic properties i De nition MXt Ee X7 provided expectation exists Note MX0 E 1 Discrete case MXt Em etmem Continuous case MXt 0700 etmemdz ii Moments Differentiating EXetX M t EX25tX M 0 EX2 In general 1190 EX Although this is basis of name 77mgf 7 it is not often useful in practice there are easier waysl iii Uniqueness Mgfs are unique That is7 if MXt My 25 for all t in an open interval containing 07 then X and Y have the same distribution This is useful as we will see below 172 Examples of mgf7s Discrete for convenience7 write 2 5t Binomial Binnp q 17p EZ0 Z P2kq k q 192 Poisson 730Iu 2200 e f uzkkl expuzi 1 Geometric Geop 2201 qk lpzk pzliqz Negative binomial NegBr7 p Elt2Xgt 224 1 qszk ltsz 2224 173 Examples of mgf7s Continuous Exponential EetX A exp77 tz dz 7 if provided if lt Gamma Ca kr71 k Mk 292707qu we AWmowfzwlexpeoewasdz was Normal NO1 note 7z2 tm 7 7 t2 t2 EetX WM exp7z2 m dz expt2m exp7m 7 25 dz expt2 700 700 174 Mgf of linear functions and sums of independent rVs i Let Y aX b Mm EexpaX bt ethexpatX ethXat ii Let X N Nuz727 so X aZ u where Z N N017 so MXt exput aztz iii Let X and Y be independent random variables W X Y MWt EexpX Yt EexpXt expYt EeXtEeYt MXtMyt iv Let X17 Xn be iid with same dsn as X W Z X Mwlttgt 2 Mat MX75 175 Immediate conclusionsH Sum of independent Binomials same p is Binomial Sum of independent Poisson any means is Poisson Sum of independent Geometrics same p is Negative Binomial and of NegBin is also NegBin Sum of independent Exponentials same rate is Gamma and of Gamma is also Gamma Sum of independent Normals any meanvariance is Normal hence any linear combination also Normal Lecture 1 Jan 5 Review of random variables Ross 42 45 51 52 11 De nitions i De nition A random variable X is a real valued function on the sample space ii De nition A random variable X is discrete if it can take only a discrete set of values iii De nition A continuous random variable X is one that takes values in foo 00 That is in principle In practice some values may be impossible 12 Examples i Discrete nite the number of heads in 10 tosses of a fair coin ii Discrete countable the number of traf c accidents in a large city in a year iii Continuous bounded range A random number between a and b values in the interval a b iv Continuous unbounded range The waiting time until the bus arrives values in 0 00 13 Probability mass function pmf or density pdf i De nition The probability mass function pmf of a discrete random variable X is the set of probabil ities PX z for each of the values z E X that X can take ii PX z 2 0 for each x E X and Ex PX z 1where the sum is over all z E X iii De nition The probability density function pdf of a continuous random variable X is a non negative function fX de ned for all values z in foo 00 such that for any subset B for which X E B is an event PX e B fXm dz B iv X takes some value in foo 00 so 00 1 Pioo lt X lt oo dm 700 14 Examples 10 1 Example 1 Binomial B10 PX m 7 35 1210 for m 01210 ii Example ii Poisson Pou mean u PX z exp7uumzl for z O 1 234 iii Example iii Uniform pdf fXm 1 for a g x g b and fXm 0 otherwise b e a iv Example iv Exponentidlpdf fXltmgt Aexpim for m 2 0 and fXltmgt 0 if m lt 039 15 Expectations of functions of random variables i Discrete case Ross 43 If X is discrete with pmf PX z pXz gt 0 for z E X the expected udlue of X denoted EX is EX EweY z pXz provided this sum exists and is nite ii Continuous case Ross 52 If X is continuous with pdf fXz fXm 2 0 for foo lt z lt 00 the expected udlue of X denoted EX is EX 30 mez dm provided this integral exists and is nite Note fXm dz z Pm lt X mdz iii Functions of a random variable EgX Emgxpxm discrete or EgX dz continuous iv Variance lf EX u varX EX 7 u2 In fact varX EXZ 7 Note varX 2 O Lecture 2 Jan 7 Review of Continuous random variables Ross 53 5 21 The probability density function de nition and basic properties i For a subset of the real line B PX6B Emma ii In fact events can be made up of unions and intersections of intervals of the form a b b Pa lt X g b fXx dx 1 iii Note the value at the boundary does not matter I PX a fXx dx 0 for any continuous random variable 1 iv Note fXx 0 is possible for some x Values see the pmf For example if X 2 0 as in the waiting time example x 0 if x lt 0 22 The cumulative distribution function of X is P7oo lt X g x The cdf is de ned for any random variable but it is most useful for continuous random variables In this case 1 d FXx m2 dz and fXx d qu 00 x 23 The Uniform distribution on ab x b l for a g x g b and x 0 otherwise 7 a lfa 0 and b 1 fXx 1 and x on 0 lt x lt1 EX12VarX112 24 The exponential distribution with rate parameter x exp7x for x 2 0 and x 0 if x lt O 17 exp7x for x gt O EX1VarX 12 25 The Normal distribution with mean u and variance 02 W 1 fXW WGXPV 2702 7 00 lt x lt oo EX u varX 02 le N Nu02 then X 7 ua N N01 26 Location and scale i A location parameter a shifts a probability density the pdf is a function of x 7 a For example we can shift a uniform U0 1 pdf to a uniform Ua a 1 pdf If X N U01 Y a X N Ua a 1 ii A scale parameter stretches or shrinks a probability density For example to transform a Uniform U0 1 density to a Uniform Ua b we shift by a and scale by b 7 a If X N U01 Y a b 7 aX N Ua 1 iii The parameter 1 of an exponential random variable is also a scale parameter le N then X N 51 le N then Y kX N k iv A Normal random variable has both location u and scale a le N Nu02 then X7lua N N01 le N Nu 0392 then Y aXb N Nau ba202 Lecture 3 Jan 9 Review of the Bernoulli process 31 The process 0 010 0 0110 010 0 0 0 0 0 010 010 01Eachtrialissuccess1ornot0 X1X2X3X4 X25 Each X is 0 or 1 T5 T10 T15 T20 T25quot Tn X1 X 7 7 7Y1 7 7 7 7Y2Y3 7 7Y4 7 7 7 7 7 7 7 7 7 Y5 7 7Y6 7 7 7 Y7 Y is rth inter arrival time W2W3 W4 W6 W7 WT is total waiting time to rth 1 The Bernoulli process is de ned by Xi independent with PXl 1 p and PXl 0 1 7 p Tn X1 1 Xn is number of successes ie 1s in rst 71 trials Y is the inter arrival time number of trials from r 7 1th success to r th WT Y1 1 Y2 K is number of trials to r th success Y Y 7 1 number of failures 0 before next success W Yf 1 Y3 number of failures 0 before r th success Note WT gt n if and only if Tn lt r 32 Bernoulli and Binomial random variables Ross 46 i X is Bernoullip PX 1 p PX 0 17p EXi px1 17pXo p p X12 1 17p X 02 p so varX EX2 7 p7p2 p17p Tn X1 1 Xn is Binomial 7119 The probability of each sequence of k 1 s and n 7 k 0 s is pk1 7p 71 PTn k k pk 71W Expectations always add ETn EX1 EXn p 19 1 p np In general variances do NOT add but here they do varTn varX1 varXn np1 7p k n and there are k such sequences 33 Geometric and Negative Binomial random variables Ross 481 482 YT are independent and have Geometric p distribution PY k 1 7 pk 1p for k 1 23 E0 Eiilk17pk 1pp117192 111 WW 17pp2 Y Y71 PY 17pkp for k 0123 EY 7 1 17 pp varY varY Recall EaY b aEY b and varaY b azvarY WT Y1 K Expectations add so EWT rp Again the variances do add varWT T17pp2 k 7 1 PWT k Pr71 successes in 71 trials and then success 17pk4prilp fOr k T7T17 T 7 1 W WT 7 r EWT 7 r varW varWT k 7 1 PWT Pr 7 1 successes in r k 7 1 trials and then success T 1 17pkp7 1p T 7 for k 0 1 23 34 A reminder about the hypergeometric distribution Ross 483 Example the number of red sh in sampling 71 sh without replacement from a pond in which there are N sh of which m are red PXm m Nim j form max0mn7Nminmn 717 Lecture 4 Jan 12 Introduction to the Poisson process Ross 47 91 41 The process Events occur randomly and independently in time at rate More formally the numbers of events N in disjoint time intervals are independent and the probability distribution of the number of events N in an interval depends only on its length 6 Additionally PNh 1 h 0h PNh Z 2 0h 42 The waiting time T to an event The waiting time T to an event is gt s if there are no events in 0 8 That is PT gt s PNs 0 E P0s P0sh P0sxP0h P0s17h7oh P0s h 7 1303 7hP0s 0h dPoPo 7de or 109130 7A3 with P00 1 So PT gt s P0s exp7s So FTs PT s 17PTgts 17exp7s So fTs exp7s on 0ltsltoo That is regardless of where we start waiting the waiting time to an event is exponential with rate parameter Recall the forgetting property77 of the exponential PT gt t slT gt t PT gt s 43 The number of events Ns in a time interval length 3 Let Ns be the number of events in interval 0 s and Pns PNs Note from 42 P0s PT gt s exp7s Then Pns h Pns17 h 7 0h Pn1sh 0h 0h PMS h 7 PMS hPn1s 7 Pns 0h Ps APn1s 7 Pns letting h 7 0 Hence from P0s exp7s we could determine P1 P2 Instead consider qns exp7ss nl Then qu exp7s ns 1nl 7 exp7ss nl qn1s 7 qns That is Pns E qns That is Ns is a Poisson random variable with mean 3 44 The conditional distribution of times of events Suppose we know exactly 1 event occurred in 0 8 At what time T did it occur This is a continuous random variable PT t 0 for every 25 Instead consider the cdf PT t FTt PT t l Ns 1 PT gt Ns1PNs 1 P1tP0s7tP1s Atexp7texp7s7tsexp7s ts So FTt ts on0lttlt s or fTt1s0lttlt 3 That is T is uniform on the interval 0 s Lecture 7 Jan 21 Linear transformations of continuous random variables 71 Location and scale again Let X have pdf fXw EX u varX lt72 and Y aX b X Y 7 ba a Location a 1 Y X b EY u b varY lt72 Fyy PY S y PX S 37 b FX3 7 b 50 fY3 fX37 5 b Scale b 0 a gt O Y aX EY up varY 1202 Fy3 130 S y PX S ya Fxya SO fy3 1afx3agt c Location and scale a gt O Y aX b EY up b varY 1202 FY3 PYS3gt PX 37ba FXy7bgtagt Sofa31gt 1afxy7ba d General case any a b Y aX b EY up b varY 1202 Suppose alt 0 Fy3gt 130 S31 PX 2 37ba P X gt 37ba 17Fx37 bgtagt X18 a continuous FV SO fY3 7 1afX3 7 Wu 1afx3 7 WW 72 Normal random variables have location and scale Recall X N Nu02 has pdf fXw 1x2 7r1aexp712w 7 ua2 So fXw 1afzw7 ua where fzz eXp71222 is pdf of N01 Recall X N Nu02 gives Z X7ua N NO1 ConverselyZ NN01 XuaZ N Nu02 Now let Y aXb auaZ b alub aaZ SO Y N Nau ba202 73 Exponential and Gamma random variables Note that exponential and Gamma densities have scale A l where A is the rate parameter Instead of a density 1afya with scale parameter a we have a density of form Afy lfY N A then AY N 51 lf buses come at rate 01 per minute they come at rate 60 X 01 6 per hour My expected waiting time is 10 minutes or 1060 16 hours lfY N Ga then AY N Ca 1 Also note I can add independent Gamma s of the same scale Tn N Gn is waiting time to nth event in Poisson process Tm N Cm A is waiting time to nth event in Poisson process so Tn Tm is just time to m nth event ie Cm n A 74 Uniform random variables Uniform random variables Ua b have both location and scale linear functions of uniforms are uniform Let X be UO1 fXu 1 for O S u S 1 and 0 otherwise Note EX12varX112 Or we could write fXu 7 u where 1 w 2 0 and 0 otherwise Note O for u S 0 u for O S u S 1 and 1 for u 21 NowletY ck7ctherecltk OSXSlsocSYSk Considercgygk Then FY3gt PY S 3 P0 k 7 CgtX S 3 PX S 3 7 Cgtk 7 0 37 Cgtk 7 Cgt So fyy1k 7 c if c S y S k and 0 otherwise 0r fY3gt H3 7 Cgtk 7 0H17 3 7 Cgtk 7 Cgtgtk 7 C 109y 7 CgtltT where 0 k 7 C is scale and c is location as required Lecture 8 Jan 23 Examples with linear transformations 81 Normal examples a Let Z N NO1 What are the mean variance and pdf of X u 0Z7 b Let X N Nu02 De ne a linear function of X that is N01 c Let Y aX b What are the mean variance and pdf of Y De ne a linear function of Y that is NO1 d De ne a linear function of Z that is N2 32 De ne a linear function of X that is N232 De ne a linear function of Y that is N232 82 Exponential and Gamma examples a If X N so show Y 6X N awe a gt 0 What is EY b Let Y1 N G314 and Y2 N G514 and Y1 Y2 independent Show Y1 Y2 is G814 Hint recall Gamma s are sum of independent exponentials c Let Y1 N G314 and Y2 N G514 What are EY1 and 7 What is the distribution of 14Y17 What is the distribution of 6Y2 d Let Y1 N 03 10 and Y2 N G54 What is the distribution of 5Y1 2Y2 Write the resulting Gamma rv as a constant times a Gamma with rate parameter 1 83 Uniform examples Let U be uniform U48 a What are EU and varU 7 b What are the distribution mean and variance of 3U 7 20 c What are the distribution mean and variance of 12 7 3U d Find a linear function of U that is uniform UO1 e Find a different linear function of U that is uniform UO1 84 Back to Normal examples Let Z N N01 a Let U 2Z 1 What are EU varU EU2 b Let V 73Z 5 What are EV varV EV27 c Compute EUV and compare it with EU gtlt Lecture 9 Jan 26 Introduction to joint distributions Ross 61 91 Joint and marginal cumulative distribution functions For two random variables X and Y the joint cclf is FX ya b PX S a Y S b 7 00 lt ab lt 00 Note that the marginal cdfs of X and of Y are given by Fxa PX a PX a YltOO Pblim PX aY b H00 lim PX a be lim FXyab E FXyaOO bANXJ bANXJ Fyb PY S b JLHgOFX yab E FX yOOb Just as in 1 dimension we can get all other probabilities from FXy For example see picture Pa1 lt X S a2 bi lt Y S 52 Pkg12752 7 Pkg11752 7 Pkg12751 Pkg11751 92 Joint and marginal probability mass functions If X and Y are discrete random variables the joint pmf is jugany PX wy y for w E X y E 3 Then the marginal pmjs of X and of Y are pxw PX w Ema0631 and py3 Z PXYC 73gt 1163 IEX Note pXw gt O for w E X and pyy gt O for y E 3 but pxywy can be 0 for some at E X y E 3 93 The multinomial distribution If we take a sample size n with replacement and there are r types with type i having probability pi i1r and 2113239 1 Let Xi be the number of type i in the sample 1 r L1 Xi n n PX1 my X2 nzwai m pl li hw n1 Recall Ross Ch 15 number of ways of arranging n1 objects type1 n2 objects type2 nk objects typek where m n2 nk n is nIn11n21nkl Example There are 4 ABO blood types A B AB and O For the USA population roughly PA 036 PB 020 PAB 008 and PO 036 Twelve students go to donate blood what is the probability 5 are type A 2 are type B one is AB and 4 are type 0 Answer 12 m036502200810364 914760 X 3257 0297 94 Joint and marginal probability density functions i Random variables X and Y are jointly continuous if there is a function fX ywy de ned for all real so and 3 such that for every set C in 892 PX Y E C f ch fxywy doc dy Then fX ywy is the joint pdf of X and Y 11gt FXyltabgt M 6 com Y 6 wow b a fxyltw7ygt dw dy 700 1700 2 8 SO fX yab mFx y z j fxywygtdw dy fXwdw iii PltXEAgt PX AY 7 Olgt uxX XA wherefXoc 1fXYw73gtd3 So fXw is pdf of X and similarly pdf of Y is fyy ff fxywydw

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