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Statistical Theory I

by: Jordane Kemmer

Statistical Theory I ST 521

Jordane Kemmer
GPA 3.79

Sujit Ghosh

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Sujit Ghosh
Class Notes
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This 3 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 521 at North Carolina State University taught by Sujit Ghosh in Fall. Since its upload, it has received 18 views. For similar materials see /class/223937/st-521-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15
1 Suppose X1X2 3 ST 521 Practice Problems U Practice Exercises for Final Exam ST 521 Statistical Theory I The ACTUAL exam will consists of lei number of problems Xn 31 fX where fXz is the pdf of a random variable X A D V Show that U FXX U0 1 where foo fXtdt De ne UFXX fort12 UnU0 1 Let X1X2 XW denote the order statistics corresponding to the ran dom sample of size n from De ne U FXX for t 12 n Show that U1 Un Show that FUW PrUn S u u if 0 lt u lt 1 and hence show that Um Betan 1 Obtain the mean and variance of Um iid A 7 V n Show that U1 U2 A O V Un are the order statistics corresponding to U1 A CL A D V Use c and d to obtain the cdf and pdf of XW Show that EXn nfol F 1uu 1du where F 1u inf 2 u Suppose X miep Nt 7 322 for t 1 5 Use Xs to construct a statistic with the indicated distribution Page 1 Sujz39t Ghosh NCSU Statistics 3 Let X7 Y be a pair of ransom variables with nite variances a Show that 91 EYlX x minimizes EY7gX2 over all functions 9a such that E92X lt 00 b Let E Y 7 X Show that i EH 0 and Vare E VarlYlXH ii CooX6 0 and more generally7 iii 6 is uncorrelated with any function gX such that E92X lt 00 c Show that VarY Vare Var X in above a is known as the least squares regression function 4 Suppose X and Y are independent random variables with nite variances a Show that CorrlX7Y 0 b Show that Corrleml 0 c Show that Corrl2X 7 3Y73X 7 2Y gt 0 d Show that CorrlQX 3Y73X 7 2Y 0 if VarlX VarlY e Show that CorrgXhY 07 provided gX and hY have nite vari ances 5 Suppose X and Y are uncorrelated random variables with nite variances a Are X and Y necessarily independent b If the joint distribution of X7 Y is a bivariate norrnal distribution7 show that X and Y are independent c Given an example of a pair of uncorrelated random variables that are not independent d Suppose U1 a1 12X and U2 b1 ng Show that CorrU1U2 0 Find a17a27b17b2 in terms of means and variances of X and Y such that EU1 EU2 0 and VarlUll VarlUZ 1 e Show that clUl CgUg is positively correlated with U1 i 01 gt 0 ST 521 Practice Problems U Page 2 Sujit Ghosh NCSU Statistics 6 Suppose that X1X2 l Z z 1131 fxlz ie X1 and X2 are conditionally indepen dent and identically distributed ciid random variables given the random variable Z A D V Find the unconditional joint density of X1X2 A 7 V Show that X1 and X2 are identically distributed A O V Assume that Varlel is nite Compute CorrlX1X2 and conclude that X1 and X2 are always positively correlated d Can X1 and X2 be independent e Can X1 and X2 be perfectly correlated ie CorrX1X2 1 7 Suppose U1 Un 31 U0 1 a Show that 72 logU1 xg b Obtain the pdf of the geometric mean H2 U01 c Let U maxlgign U and Um minlgign U be the maximum and mini mum order statistics Obtain the joint pdf of U1Un d Compute CorrU1 UM e Obtain the pdf of the range R Um 7 U0 f Compute ER and VarlR 8 Let X be a positive valued random variable with nite variance Show that a VarllogX lt 00 and VarlxX lt 00 b CorrllogXX 2 0 c COTT X S 0 d S eE10gXl S When does the equality holds this is the famous inequality HM 3 GM 3 AM 9 Give an example of a pdf a that belongs to both exponential family and location and scale family b that belongs to an exponential family but not to a location and scale family c that belongs to a location and scale family but not to an exponential family 10 Review 21711 the problems solved during Lab hours ST 521L and in PP l ST 521 Practice Problems U Page 5 Sujit Ghosh NCSU Statistics


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