REGRESSION ANALYSIS M 384G
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This 5 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 49 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
CONDITIONAL AND MARGINAL MEANS M 384G374G 1 Question How are conditional means EY IX and marginal means EY related Simple example Population consisting of 111 men 112 women Y 2 height X 2 sex Categorical two values Male Female So there are two conditional means EYlmale 2 Sum of all men s hei ghts r11 EYlfemale 2 Sum of all women s heightsn2 Then Sum of all men s heights 2 n1EYmale Sum of all women s heights r12 EYI female The marginal mean is EY 2 Sum of all heightsn1 n2 2 Sum of men39 s heights Sum of women39s heights quot1 quot2 n1EY male n2EY female n1 n2 n2 EYI female quot1 EYI male nl n2 n1 n2 2 proportion of males EYI male proportion of females EYI female 2 probability of male EYI male probability of female EYI female Thus The marginal mean is the weighted average of the conditional means with weights equal to the probability of being in the subgroup determined by the corresponding value of the conditioning variable Similar calculations show If we have a population made up of m subpopulations popl pop2 popm equivalently if we are conditioning on a categorical variable with m values eg the age of a fish then EY E Prooa EYI pop k1 eg for our fish popk 2 sh of age k and 6 Elength E PrAge kE Length I Age k kl Rephrasing in terms of the categorical variable X defining the subpopulations EY EPrxEY I X x all valuesx of X Stated in words The analogue for conditioning on a continuous variable X is EY ffXxEYxdx where fxx is the probability density function pdf of X Note 1 There are analogous results for conditioning on more than one variable 2 The analogous result for sample means is y 11 A second related relationship between marginal and conditional means for populations Consider EY IX as a new random variable U as follows Randomly pick an X from the distribution of X The new rv U has value EYIX x Example Y 2 height X 2 sex Randomly pick a person from the population in question U W EY X female if the person is female Mm E Y IX male if the person is male Question to think about What might cause PU u to be high Hint There are two ways this might arise Consider the expected value of this new random variable e g the expected value of the mean height for the sex of a randomly selected person from the given population In this case we would expect EU to depend on the proportion of the population which is of each sex If U is discrete then EU 2Puu Why All poxxible mum of U Example For U Eheight sex the values taken on by U are and with respective probabilities and so EU which from Part I is just In other words EEhtsex The same reasoning works in general showing that The expected value of the conditional means is the weighted average of the conditional means marginal mean which from Part 1 is just the marginal mean ie EEYIX 2 weighted average of conditional means Y M 384G374G JOINT MARGINAL AND CONDITIONAL DISTRIBUTIONS Joint and Marginal Distributions Suppose the random variables X and Y have joint probability density function pdf fXYYXy The value of the cumulative distribution function FYy of Y at c is then FYc P Y S c PooltXltooYSc the volume under the graph of fX7YXy above the region quothalf planequot oo lt x lt oo Sketch the reglon and volume yourself y S 0 Setting up the integral to give this area we get me JJ fXYxydxdy I I may dx dy mego dy where gm IfXYxydx 5 Thus the pdf of Y is fYy FY y gy In other words the marginal pdf of Y is my IZfXYxydx Similarly the marginal pdf of X is fax IZgXYxydy 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 pdf s or pmf39s probability mass functions if you prefer this terminology for discrete random variables are de ned by fYy PY y and fXX PX X The joint pdf is similarly fxyXy PX X and Y y The conditional pdf of the conditional distribution Y X is fv xYX PY YX X PXxandYy PX x Jimmy fXOC I Is this also true for continuous X and Y In other words IS Idexaay fXW PcSYSdXaforeverya a y PY S d X a for every a Draw a picture to a It is enough to show that Id fX39Y fX help see why Starting with the right side we can reason as follows Draw pictures to help see the steps PYSdXazPYSdaSXSaAX forsmallAX PYSdandaSXScHAx PaSXSaAx H PYSdandaSXScHAx fXaAx
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