TOPICS IN STATISTICS
TOPICS IN STATISTICS MATH 218
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This 2 page Class Notes was uploaded by Donald Gusikowski on Monday October 5, 2015. The Class Notes belongs to MATH 218 at Clark University taught by David Joyce in Fall. Since its upload, it has received 23 views. For similar materials see /class/219538/math-218-clark-university in Mathematics (M) at Clark University.
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Date Created: 10/05/15
Summary of basic probability theory part 3 D Joyce Clark University Math 218 Mathematical Statistics Jan 2008 Joint distributions When studing two related for Y We can write these marginal functions on real random variables X and Y it is not enough the margins of the table just to know the distributions of each Rather the pair X Y has a joint distribution You can think Y fX of X Y as naming a single random variable that 1 0 1 2 takes values in the plane R2 1 0 136 16 112 518 X 0 118 0 118 0 19 1 2 Joint and marginal probability mass func tions Let7s consider the discrete case rst where both X and Y are discrete random variables The probability mass function for X is fXx PX7 and the p39m39f39 for Y ls fYy PYy39 Discrete random variables X and Y are indepen The 30th random variable X7 Y has its own p39m39f39 dent if and only if the joint pmf is the product of denoted fXyzy or more brie y fxy 0 136 16 112 518 112 0 112 16 13 fy 536 118 1736 13 the marginal pmfs u fx96fyy In the example above X and Y aren7t independent fX Z my fyy Z Ly Joint and marginal cumulative distribu y m tion functions Besides the pmfs there are The individual pmfs are usually called marginal 30th and marginal cumUIatiVe diStribumon fumequot pmbabilZty mass functwns39 tions The cdf for X is PX x while For example sssume that the random variables the C39d39f39 ff Y 13 Fyy 13039 The 301m X and Y have the joint probability mass function random variable X Y has its own cdf denoted given in this table FXgtYxvyv or more brle y Fay may PXYzy PX96 and Yy and it determines the two individual pmfs by Fy 13ng and ng 16 112 and it determines the two marginal pmfs by X 0 1 18 0 1 18 0 1 0 136 1 6 112 Fxltzgt yhggo Fwy my 1111 my 2 112 0 112 16 J01nt and marginal probability den51ty By adding the entries row by row we nd the the functions Now lets consider the continuous case marginal function for X and by adding the entries where X and Y are both continuous The last column by column we nd the marginal function paragraph on cdfs still applies but we7ll have 1 s Furthermore the marginal density functions can be found by integrating joint density function amfama7hmfamm Continuous random variables X and Y are inde pendent if and only if the joint density function is the product of the marginal density functions u fx96fyy Covariance and correlation The covariance of two random variables X and Y is de ned as COVX7Y UXY EX MxY MY lt can be shown that CovX Y EXY 7 any When X and Y are independent then oXy 0 but in any case VarX Y VarX 2 CovX Y VarY Covariance is a bilinear operator which means it is linear in each coordinate CovX1 X2Y CovX1 Y CovX2Y CovaX Y a CovX Y CovX Y1 Y2 CovX Y1 CovX Y2 CovX bY b CovX Y The correlation or correlation coe cient of X and Y is de ned as UXY PXY UXUY Correlation is always a number between 71 and 1
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