Question

Consider the binomial distribution with parameters n 10 and p (unknown) a) Is this binomial distribution an exponential family distribution? b) Find a sufficient statistic for p.

Answer 1

A family of pdf or pmf is called an exponential family if it can be expressed as a f( 0) = h (2) CC) lxp ( I W;(O) ti(n)) ahore near zo, tica) are real valued funct. e que observation &.ccorzo, wico) are also real valued fune of the vecton valued poorameter Opci The pmf of Bironial cio, p) is - ſcalp) = ( ) px (1-P) '-, 200,..40 a) cico)" (ir)* -() Co-p}'° exp [1024 x log(n+7 ushere, n(a)= (),2=0,1...10 o oiw CCP=(1-'', W.(n) = log litt t,(a) * :. flarplan(8) ect exp (t, (a) W, (PD) By definition Binomial désam with n 210 . f unknown p is an exponential family distribution. 6 det, y be a random variable having pmfon poff (). A stal'stie trou is sufficient for o is the distribution of P x given xl tex) is independent of a le $ (x/t,0) b (Wit). By Heyman - fisner factorization theorem the statistic tas in sufficient for o iff the

and density can be factorized as h(x) buto a tune of parameter on a through tean o but only depending on die f(x 10) = app n (a) b12,0) Now, $ (alt) = $18!) ☆ (H) (19) DP (1-0) 0 (8) = (1-P). 6 = (1-1)!" ( ) lie. x is sufficient for p .. where t zx. 1 - 10