Question

Bernoulli distribution with parameter θ .

a) Use the method of moments to obtain an estimator of θ

b) Obtain the maximum likelihood estimator (MLE) of θ.

Answer 1

pmf. fex) = at 1-7 2-01 Llola wala Leta x be a bernoulle variable with para meter o. i.e X ~ Berlon24 texa, or Co-os a). Now, By Method of Moment f. Mom). method mean (2) with Expectation of we need to equate sample E(x) = 50.610) Loco E (X) = Ia We consider a sample from above pool b) Let, we have a sample ine. X1, X2 Xn Berto) Joint pmf, for MLE Likelihood as ليه دي So, И n 12 n-exi IX 0 We n-exil Claim Now, taking Logarithm we get d=an Louan) = {xilno + (n-2x;) ln (1-2) Now, derivative w.rt we get, axi in-sx we will equate to Zero for MLE, 2 X" o 3 x _ na zxi OML 1-9 n