Mathematical Statistics I PDF

Missing Thursday notes Normal Approximation to Binomial...

Missing Thursday notes Normal Approximation to Binomial O page 10s tru gamma (Y)p x B(n , p) + p)n - EX 3 3 2 pg 105 y ( , p(x) · - = -.... nur X ~Bin (n = 25p = 0. 6) : n(X) 13-K (p P(x113) [ (1)0 6kx0 4 0 267 = = =... K 0 S p* = ~ using normal approximation · want to exp gas = N(15 6) , , where approximate · EX = np = 25x0. 6 = 15 · VarX = np(1 - p) = 15x0. 4 = c * try poisson · inen , Plz Example where is not an exponential family P(z = 82) X ~un(0, f(x) [(0 X has a better approximation. log(1-pP 1 - * In general-p(X = X) = p(X = X +0. 5) = wp = 10g(p) = log(1 p) - = =p + - Tp - P (N(np p)) (ep X) =p. = np(1 n = - =. , E * np = 5 · p(X = X) = P(X = X-0 5). E(X) = n - p n(1 p)15 P(N(np np(1 p)) - = - , · Exponential Family · f(x10) = n(X)c(0) exp[ti(x) - , X support does not depend on a EX. XwEXp(x) , f(x(x) = Je * 5 exp) Y x) = - - a n(x) = - =X. X-N(U , 0) , f(x 14 , 0-5" expl- (X= = - 24x + y2)3 > - = expl -02X2 + X) - c(y , 5) Location and scale family e.. Let9(x) g = (*) Location P( 0. P(q(X)[r) 1 # : P(g(X) Ir) = S 1) fx (x)dx = + (g(x)f(x)dX (X g(x)2r) : (X g(x)2r) : - Eg(X) =g(x) = z(X -4x)2A(X-EX) = var  Bivariate distribution (X , Y) X jf(xax Siexdx 1 1 · Discrete = = 3 Sty() 01 2 total ① dy = S. 3y2 1 = O 0 1. 0 2. O g 0. 3 T O 05 0 0 5 Y 0 0 25 marginal... 2 0 o 05 0 05 0 025 0 125 put for Y.... z O ⑧. 0 025 0. 05 0. 075 total 0. 3 0. 5 0. 125 0. 05 1 Il Il II Il =O D =PIX y) X][P(X p(X x) y [P(X X = p(x) + 0+ y) = = = · = = , , pm , P(y = P(X y p(y) y) X y) Y · = = put = = , , of = 4 0 S X 0 = = 1 P(y) = 2 ins = 0. 125 y = 2 everything 0. 075 = 3 0 025. y = 3 * know now to calculate marginal mean Continuous : FC) Area under the curve · P(X += 1)] Ex , y (x, 4) ayay S(tx y Y)dxdy -Size vomme (X / = , N -Stea 3 "IIIIIIIIII => = 1 C'yz/y - 0 Y11 A fx y (x < Gxy2 = Ex · , , y) = , 0x T x - 1 = · fx (x) J: exy2 dy = x dy · = oX3 2x 3((2y3 = fy(y) -Sooxy2dX yz)dy - = = = 42 zy 3[ - ) = = = 3[' -]