Mathematical Statistics MT 427
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This 8 page Class Notes was uploaded by Mr. Halie Wilkinson on Saturday October 3, 2015. The Class Notes belongs to MT 427 at Boston College taught by Staff in Fall. Since its upload, it has received 26 views. For similar materials see /class/218068/mt-427-boston-college in Mathematics (M) at Boston College.
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Date Created: 10/03/15
Probability the good parts version I Random variables and their distributions continuous random variables A random variable rv X is continuous if its distribution is given by a probability density function pdf ne that is positive on an interval For real numbers a lt b Pa lt X lt b bfacd Random variables X and Y are jointly continuous if there7s a joint den sity function fxy such that b d PaltXltbcltYltdfxydydac The marginal density for X is found by integrating out the y and vice versa for Y X and Y are called independent if their joint density is the product of their marginals In this case it follows that Pa lt X lt bc lt YltdPaltXltbPcltYltd The cumulative distribution function cdf of X is the function F Fa PX ac ftdt Given a function 90 the expected value of gX In particular the mean of X EX M the variance of X VX 02 EX 7 m2 EX2 7 M2 and the standard deviation 039 0392 The moment generating function of X is Mm EM ewtfmdyc Properties of the moment generating function 1 M0 Mxt evaluated at t0 EX 2 1f X1X2 X71 are independent random variables then MX1X2Xnt MX1t gtk MX2t gtk gtk MXnt 3 The mgf specifies the distribution if MXt Myt then X and Y have the same distribution 4 MaXbt Cthxat The standard class of continuous distributions lt1 X No02 density fac ime w HVZUQ foo lt x lt 00 mean M variance VX 02 mgf Mt e t 72t22 If X N NM02 then X 7 M0 N0 1 If X N NMXU and Y N NMy032 are independent then 1X 191 NapX 1 buy 120 1 192033 2 X N Chi Square With n degrees of freedom X N X2n if X N Gamman212 mean n variance VX 2n mgf Mt WZJ lt 12 3 X N EXponentialA density at Ae m c 2 0 mean lA variance VX lA2 mgf M05 t lt A PX gt x 6 for x gt 0 Note There are 2 common conventions followed for the EXponentialA i A is the parameter in the density as above and the mean is lA and ii A is the mean and the density is at lAe m the interpretation in a particular problem should be clear from context 4 X N GammaozA density at Egaca le mw 2 0 mean aA variance VX ozAZ mgf M05 f lt A Note Na fooo ta le tdt PM n 7 1 When n is a positive integer When X1X2 Xn EXponentialA are independent X1 X2 Xn Gamman A In particular EXponentialA Gammal A 5 X N Ua b density at lb 7 a for a lt x lt b mean a b2 variance VX b 7 1212 bt t mgf M05 5b 700 lt t lt oo lf cd C ab then Pc S X S d is LengthcdLengthab II Random variables and their distributions discrete random variables A discrete rv X takes on a nite or countable number of values Prob abilities are computed using a frequency function pk PX k this is also called a probability density function pdf or probability mass function PaltXltb Z pk k ab Given a function g the expected value of gX E9X 2 9001906 k in particular the moment generating function of X is Mxlttgt ElteXtgt Zektpc k The standard class of discrete distributions 1 X N Bernoullip frequency function p1 pp0 q 1 7 p mean p variance VX pq mgf Mt qpet 700 lt t lt 00 2 X N Binomialnp frequency function pk pkqn k k 01n mean np variance VX npq Ingf M05 qpet foo lt t lt 00 If X is the number of successes in n independent Bernoulli trials then X N Binornialnp 3 X N Ceornetricp There are two de nitions for the Ceornetricp distribution l X is the number of failures required to see the first success in a sequence of Bernoulli trials and H X is the number of trials required to see the first success in a sequence of Bernoulli trials If X and Y represent have these respective distributions then Y X 1 We give results separately for the two definitions Case I frequency function Mk qu k 01 where q l 7 p mean qp variance VX qpZ Ingf M05 1 foo lt t lt lnlq Case ll frequency function Mk qu l k 12 where q l 7 p mean lp variance VX qp2 t Ingf M05 1Zenioo lt t lt lnlq 4 X N Negative Binornial7 p Again there are two definitions for the Negative Binornial7 p distribution l X is the number of failures before the rth success in a sequence of Bernoulli trials and H Y is the number of trials required to see the rth success in a sequence of Bernoulli trials If X and Y have these respective distributions then Y X 7 We give results separately for the two definitions Case I kr71 frequency function pXk lt r71 pqu s 01 Where q 17p mean rqp variance VX TqpZ mgfi Mt1et eoo lttlt1n1q Case II kil frequency function pyk lt 1pqu fk 737 1 Where 7n q1p mean rp variance VX rqpZ Inng Mos 157392 700 lttlt1n1q 5 X N Poisson frequency function Mk 5711 16 01 mean A variance VX A Ingf M05 e et 17oo lt t lt 00 III General Properties 1900 EaX bY aEX bEY for any random variables X and Y EXY if X and Y are independent VX and 000X Y VX EKX M2l EX2 M2 VaX b a2VX VX Y VX VY 2000X Y for any random variables X and Y VX Y VX VY if X and Y are independent 00M Y 7 ElltX 7 MW 7 MYl 7 MY 7 wow CorX Y 0 if X and Y are independent CovX X CovaXbY abOovX Y CovXY U V 000X U CovXV CorY U 0011Y V Sampling lf are n independent identically distributed random variables With M and 02 and X zzl Xi is the sample mean then M and VX 0211 Central limit theorem lf are n independent identically distributed random variables With M and 02 then 2211 Xi approximately NnM 1102 as n a 00 X approximately NM02n as n a 00 Joint and conditional distributions If X and Y have joint pdf fX7yxy then mm ff fxy9c ydy or San y fXY7 y fYy ff fXY7yd OF 2am X fXY7yi fXYy fXYz yfYy leXmlty fXY7yfXi fX7yxy fX xfyy if and only if X and Y are independent Xmaxz Xmin lf are n independent identically distributed random variables With pdf fXx then fXWW nfxlFxl 1 mem nfxl1 Fxl 1
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