Question

5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let X1,..., Xn bei.i.d. Poisson random variables with parameter 1 > 0 and denote by Xn their empirical average, Xn=1 İx; and (bn), such that an (Xn-bn) converges in distribution to a standard Gaussian random variable Find two sequences (an) Z~ N(0,1). an =

Answer 1

TOPIC:Application of the Central limit theorem.

Here, X, X2, -, xn are iid Poisson bandom variables with parameter, xro. e xi Poisson (2) ; & 1 = 1,2,- M . Defina, on - Empirical Average. I we know, when, X & Poisson (2); then, E (xi) = 1, var (xi) = 2; &i=1,2,,n. Now, E(In) = 1 1 2 E (xi). n2 TUE (xi) = 2; 7 = 1 & i=1,2,-,n. L ГДug to and, var (3n) = var (Ex). var (x). kindependence [o! Var (x:) = 2,7 *i=12-in) =

central limit theorem; we BY geben En un EE1, . N(2,-*) → (3x - 2) (0, ). . It www, c9 L = (x-)wN(o,) (6-1) (6 ( an (In - b) u (0.1). - For two sequences (an) nee and (bn) ux - where where an = bu = a senswers bn = 21 Answers