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# REGRESSION ANALYSIS M 384G

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This 7 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 384G at University of Texas at Austin taught by Martha Smith in Fall. Since its upload, it has received 50 views. For similar materials see /class/181458/m-384g-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15

M384G CAM 384TM374G JOINT MARGINAL AND CONDITIONAL DISTRIBUTIONS Joint and Marginal Distributions Suppose the random variables X and Y have joint probability density function pdf fXVYxy The value of the cumulative distribution function FYy of Y at c is then FYc P Y s c P wltXltwY sc 2 the volume under the graph of fXVYxy above the region quothalf planequot w lt x lt 00 R Sketch the reglon and volume yourself y s 6 Setting up the integral to give this area we get W fffxyx dey R 2 f nyOC 3 dx dy mgy dy where gy 1fxvyx ydx Thus the pdf of Y is fYy FY y gy In other words the marginal pdf of Y is My fXyx ydx Similarly the marginal pdf of X is fax ffxyltxygtdy In words The marginal pdf of X is Note When X or Y is discrete the corresponding integral becomes a sum Joint and Conditional Distributions First consider the case when X and Y are both discrete Then the marginal pdfs or pmfs probability mass functions if you prefer this terminology for discrete random variables are defined by My M y and fax PX x The joint pdf is similarly fxyXy PX X and Y y The conditional pdf of the conditional distribution YX is fYXyX PY y39X X PX x and Y y PX x fXYxsy fXx 39 In words Is this also true for continuous X and Y In other words Does d fX ym y dy Pc 5 Y s d X a for every a c and d C fX a It is enough to show that fWWdy PY s d X a for every a and d Draw a X a picture to help see why Starting with the right side we can reason as follows Draw pictures to help see the steps PY sdXaPstasXsaAx forsmallAX PYs dandasXs aAx Pas Xs aAx PYs dandasXs aAx mam fj AXnyxydx dy fXaAx fifxymymxdy fxomx fin yummy Z fX a d 39 deYa y y as desired fXa Summarizing The conditional distribution YIX has pdf fXYxy fYXyX 7 x In word equations joint density of X and Y Conditional density of Y given X I I marginal densin of X and of course the symmetric equation holds for the conditional distribution of X given Y INFERENCE FOR COMPARING TWO MEANS There are two inference procedures for comparing means of two independent samples I Pooled Two Sample t Procedures These are often emphasized in introductory mathematical statistics textbooks and in older statistics books They make the assumptions that 0 A simple random sample of size n1 is taken from a normal population with unknown mean MI and an independent simple random sample of size n2 is taken from another normal population with unknown mean Hz 0 The two populations have the some unknown standard deviation 039 The test statistic used is f1 f2u1 u392 S ii p quot1 quot2 where sp is the pooled standard deviation S n 1sfn2 1s p n1n2 2 Under the assumptions of independence and equal population standard deviations this statistic has a t distribution with n1 n2 2 degrees of freedom The pooled t procedures were common before statistical computer software was available because they are relatively easy to use However there are problems involved in using them 0 The assumption of equal standard deviations is hard to verify in most cases In particular statistical tests for equal variances are not robust to departures from normality even with large sample sizes When sample sizes are quite different the pooled t procedures are not robust to unequal standard deviations Unequal standard deviations are common in real data eg data sets with large means tend to have large standard deviations M 384G374G JOINT MARGINAL AND CONDITIONAL DISTRIBUTIONS Joint and Marginal Distributions Suppose the random variables X and Y have joint probability density function pdf fXVYxy The value of the cumulative distribution function FYy of Y at c is then FYc P Y s c P wltXltwY sc 2 the volume under the graph of fXVYxy above the region quothalf planequot w lt x lt 00 R Sketch the reglon and volume yourself y s 6 Setting up the integral to give this area we get W fffxyx dey R 2 f nyOC 3 dx dy mgy dy where gy 1fxvyx ydx Thus the pdf of Y is fYy FY y gy In other words the marginal pdf of Y is My fXyx ydx Similarly the marginal pdf of X is fax ffxyltxygtdy In words The marginal pdf of X is Note When X or Y is discrete the corresponding integral becomes a sum Joint and Conditional Distributions First consider the case when X and Y are both discrete Then the marginal pdfs or pmfs probability mass functions if you prefer this terminology for discrete random variables are defined by My M y and fax PX x The joint pdf is similarly fxyXy PX X and Y y The conditional pdf of the conditional distribution YX is fYXyX PY y39X X PX x and Y y PX x fXYxsy fXx 39 In words Is this also true for continuous X and Y In other words Does d fX ym y dy Pc 5 Y s d X a for every a c and d C fX a It is enough to show that fWWdy PY s d X a for every a and d Draw a X a picture to help see why Starting with the right side we can reason as follows Draw pictures to help see the steps PY sdXaPstasXsaAx forsmallAX PYs dandasXs aAx Pas Xs aAx PYs dandasXs aAx mam fj AXnyxydx dy fXaAx fifxymymxdy fxomx fin yummy Z fX a d 39 deYa y y as desired fXa Summarizing The conditional distribution YIX has pdf fXYxy fYXyX 7 x In word equations joint density of X and Y Conditional density of Y given X I I marginal densin of X and of course the symmetric equation holds for the conditional distribution of X given Y

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