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# Introduction to Mathematical Statistics MATH 309

UW

GPA 3.8

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This 2 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 309 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/205287/math-309-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15

STATMATH 309 DISCUSSION 6 TA JingjiangJack Peng Of ce 1275 M807 1300 Universtiy Avenue Email pengstatwiscedu Phone 262 1577 Of ce Hour 1130 130 pm Tuesday or by appoitment Website wwwstatwiscedu peng Joint Distribution o It is often important to keep track of the joint probabilities of two random variables7 X and Y 0 Their joint cdf is given by Fxy7y PX S LY S y 0 If X and Y are discrete7 then their joint probability mass function is given by pr RXYW o If X and Y are continuous7 then the joint density function fXY7y 32122 33 PEG79 fix fix fxy87td8dt 0 The marginal density of X7 and Y can be computed from any Fxy7pXy7y7 fXVY O FXy7oo 00 fXyz7ydy mm3 mm3wmmm I fXgtY7yddy 1 2 Conditioning and Independence o If X and Y are discrete7 then the conditional probability mass function of Y7 given X7 max121 equals leXltyl pXW o If X and Y are continuous7 then the conditional density function of Y7 given X7 equals fxy w mm W 0 Discrete X and Y are independent if and only if gymLy pXpyy for all X and y7 or equivalently py Xylx pyy 0 Continuous X and Y are indepednet7 if and only if fxy7y fXfyy for all X7y7 or equivalently7 fy Xylz fyy 3 Examples 1 Absolutely continuous random variables X and Y have joint density function czzyz form lt1 0 otherwise Jew9671 a nd the value c7 b nd the marginal density of X7 c nd the marginal density of Y7 d nd the conditional density of Y given X057 e nd the conditional density of X given Y05 2 278 amp 2823 Let X and Y have the Bivariate Normalp17p27017027p distribution7 a Prove that X N NM17039127Y NM27Ug bSuppose that prove that Y given XX is distributed NW2 p02z 7 m7 1 7 p2a cEstablish the analogous result for the conditional distribution of X given Yy AA 3 2821 Suppose that X17X27 7X is a sample from a distribution with cff F Prove that n FX Z 7Fquotltzgtlt1 7 MW jz j hint Note that Xi S 239 if and only if at least i of X17 Xn are less than or equal to X

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