ch5&7 - Joint pmf= sum of all possible outcomes:...

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Joint pmf = sum of all possible outcomes: EX) given probability chart, P(1error)=P(0,1)+P(1,0) Ex) 2die rolled, PDF= 1/36, CDF=XY/36. Ex) Poisson = ! / x e x λ - , PDF of XY = multiply both Joint pdf = ∫ ∫ b a d c dxdy y x f ) , ( =1. Ex)f=kxy. 1<x+y<2. ∫ ∫ - - - + 1 0 2 1 2 0 2 1 x x x kxydydx kxydydx Ex) f=c/x. 27<y<x<33. ∫ ∫ 33 27 33 / y xdxdy c . P(i<30,o<28)= ∫ ∫ 28 27 30 / 72 . 1 y xdxdy Marginal PMF: just sums of columns. PDF: - = dy y x f x p x ) , ( ) ( and py(y)=dx. Ex) f=1/240. 8.5<x<10.5. 120<y<240. = 240 120 240 / 1 ) ( dy x p x , = 5 . 10 5 . 8 240 / 1 ) ( dx y p y Ex) CDF(y)? f=axy. a=1/24. 1<x<3. 2<y<4. 6 / ) ( 3 1 y xydx a y p y = = . 12 / ) 4 ( 6 / 2 2 - = = y dy y CDF y Expect. Val: E[h(x,y)]=sumxy[h*p]. for E(xy) sum everyvalue*prob. For E(x) sum margprob*rowvale Ex) f=2y. 0<x<1. 0<y<1. E(XY)= ∫ ∫ 1 0 1 0 ) 2 )( ( dxdy y xy watch xy operator Covar: Cov(x,y)=E(XY)-E(X)E(Y). 0=weak dependency Ex) f=2x, 0to1x&y. E(xy)= ∫ ∫ 1
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