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# Introduction to Mathematical Statistics MATH 309

UW

GPA 3.8

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This 3 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 16 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 5 TA JingjiangJack Peng O ice 1275 MSC7 1300 Universtiy Avenue Email peng statwiscedu Phone 262 1577 O ice Hour 1130 130 pm Tuesday or by appoitment Websit wwwstatwiscedu peng 1 Cumulative Distribution Function 0 cdf PX S s o i0 S S 1ii FXoo 17 and FX700 0 iii is non decreasing o For discrete random variables7 PX S s 2mg PX o For continuous random variable7 fix fX1tdzt7 Where fX is called probability density function 2 Onedimensional Change of Variable o If X is discrete random variable7 Y hX7 then pyy PY y PhX z PltX was Emmy PltX z hil o If X is continous7 Y hX7 then fyy W 3 Examples Y N N78747 calculate PY S 757P72 S Y S 77PX 2 3 2 6 3 X N NM7027Y 0X d7 where c gt 0 Prove Y N Ncp d70202 X N ezp7 Y X37 Compute the density fy of Y STATMATH 309 DISCUSSION 9 TA JingjiangJack Peng O ice 1275 MSC7 1300 Universtiy Avenue Email pengstatwiscedu Phone 262 1577 O ice Hour 1130 130 pm Tuesday or by appoitment Website wwwstatwiscedupeng Conditional Expectation If X is discrete7 then the conditional expectation of X7 given an event A7 is equal to EXlA ZxPX MA If X and Y are discrete7 then EXlY is a random variable7 with EXlY equal to EXlY y when Y y If X and Y are jointly continous7 then EXlY y f fx y yd7 and again EXlY is a random variable7 with EXlY equal to EXlY y when Y y Conditional expectation is linear7 ie EaX1 nglY aEX1lY bEX2lY Law of total expectation EX E9YEXY Elgmxr Emmy gm EEltXYgtm EltXm Conditional Variance formula VarXlY EX2lY 7 EXlY2 Calculate Variance based on conditioning VarX VarEXlY EVarXlY Inequalities For nonnegative X7 Markov7s lnequality says PXgeqa S Chebychev7s lnequality says POY 7 My 2 a S V Y The Cauchy Schwartz lnequality says lC oMX7 Y S VarXVarY12 So lCorMX M S l Jensen7s lnequality fEX S EfX whenever f is convex 3 Examples 1Suppose X and Y are discrete with 15 27y3 15 z37y2 15 37y3 15 z27y2 15 z37y17 0 otherwise IRA71 a Calculate EX1Y 37 EY1X 37 EX1Y7EY1X b Use the above calcuation to verify EX7 VarX VarEXlY EVarXlY 2 Suppose X li oisson17 Y N Poisson2 X and Y are independent Let Z XY7 Calculate EX1Z 3 Suppose that X7Y Bivariate NorrnalmL7 27 017 027 p Find EX1Y7 EY1X7 VarXlY7 and VarYlX 4 Let Z N li oisson37 Use Markov7s Inequality to get an upper bound on PZ 2 7

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