Y F totalexpL Ky fy g dy Ef ace Poisson y f yeidy I law of total - PDF document

r.ve's Practice with continuous Total Probability Law of Discrete 2 M D Pray PrfEI ME fPrCE Y g fy g dy Pr E Law of Total Expectation Discrete Continuous k Pr Yek E XlY F x w y fyb dy fE X Y F X E PrCx xH D Prcx Y D xf µ ylYdx w f x y y

Example wrandan a person has in a year accidents is but 2 rn is a Poisson exp l Bindooi A D instead A population that has fraction of a EEE prfnoauidfa kq.ie is g F i go Prf randomperson has no accidents 1awM meg no act 2 9 G d's 1 has Prfrandn person exppy fxcxt.ae Xnexpli X Pousson ly E P4X k y e 9dg e co e 4dg EE XR k s Efx oPr Kk k accidents for randenpenseningeay Y F totalexpL Ky fy g dy Ef ace Poisson y f yeidy I law of total probe 0 Pr Xd Pr XcY Y fyGldy Pr Xss Fxls

I X nu g X Y find busing 4 2X 1 I f Cx Fx x X probdensity fn for Y 2 4 S 25 to for computing fyG Recipe FyGy Pr Yes CDF for Y Find Prf2X Eg Pr Yes Pr g as Fyls Pr Xe nurse Efx FX differentiate to get p.d.fr fy dagFyG yFyG IzFxl lfxfaj.LT 2 X w PrfXZw Fwiw PrfrwsXerw makes sense i e to see this intuition Efx X sates a Prfa small far epsilon Y 2X E fy a age Yeates Pr a Prfa Eze 2XeatE Prfaz Eye KITE E ez.f fx E E

Xv Nlp d Y aXtb aha Yn Nfaprtb Claim Pr Yey Pr axtbey Fyly Proof Fx Ia prf xey.bz aLfx Ia dd yfxfs.bz e lEs nI taa 9 Hae N aptb aka which is density of

Joint Distributions Pr X ex Yey F x g Joint CDF y I f F dydx asb joint density fu x g f f zy Ey x y Lf I f dx dy x y fC y dydx a flap dadb Ytbtdb b Pr X Laida a Marginal density for so f x g dy f f x oo prob mass fus Conditional Independence V xD fxycx.us 1 6 ly y Y indep X y with fyly fXYk f for o y yCx f y y conditional prob densityfn f yay fy g fxpy.gl fxkHyµ

Example X Unif 0,10 10 f Xl 3 o Too ifX 3 ynunifo.MY fyniff i yoe o ow F Y any total expectation E Y X x f lx dx 10 2.5 dx f Iz fo O discreteselting f Glfyµ XRY discretenu s fxylx.us prfk x.to w x Y y Pr PrfX Uniffoyo Uniffo x PrfY y X x Pray foot