Question

Let {x1, x2, ..., xn} be a sample from Bernoulli(p). Find an unbiased estimator for p^2 .

Let {x1,x2,..., Xn} be a ..., Xn} be a sample from Bernoulli(p). Find an unbiased estimator for p?.

Answer 1

Qus: find the unbaised estimator of p^2 for Bernoulli distribution

Solution:

Solution - be a sample from Rerolls (o) We have to find an unlaised for x=0, estizator for p² We know that if X ~ Berbullillat the podelility Mare function of is given by P/x+x)=) pr (pptx i o ow Now we find Expection of Randon Variable x from defnition E(x) Ex.p(x) EX.P(x) = 0 pº (1-P)! Flip'll-p)/- lot lopil 11 RO 1-01 11 p Now ملط Elx) = p. We find Expected value of Š [7])= ( x) 63xi) - t iş 2 Elxi) 191 (1 ER 11=1 &p=p&l=hp Iyme 12 so x is unlaised cetinator for

again we know that Veriance of Randon varitte & which fallow Berboulli (f) is equal to so (where gitt Vas (Y) = P(1-P) We know that Vorld) [(x2) again we find variance of X Varli Vorl 1 2x) + Velfilepfity E 1 & varlxi) 1² 121 he P (142) 11 P(1-1) El h b 12.11 - Pli-e) b2 plice) We know that Var(8) E(72) - {E(F) 4² P(1-P) (74 - p? b (using eqh #) => ELI?) - P(1-) p2 b.

(7) - p?+ PC1-2) 12 р? + Р pe 1) 1 ܠܐ E (X² - Do = (1-1) p² + b Elx²) p. (1-4) p² (1-1) p² (p= E(8) b Ebx) = GELY) E(72) - E(1) (1-4) p? (1-1) p² Elt 3² - (3-1) p² F{ 57-72 pe 2 y 4, 6( 372 - 7) = 1² :: E(1x) = 6 6CV) Here h constant E/68²-X -p? so br² - ie Uhleised estimates of p2 B. 1-1 (67 17-1 6-1