Question

The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls received at different hours are considered to be independent. Emergency calls X1 ,…, Xn for n consecutive hours has the same parameter λ.

a) What is the distribution of Sn = ∑ Xi ?

b) Provide Normal approximation for the distribution of Sn .

c) Provide maximum likelihood estimation of λ. Calculate variance and bias of MLE.

d) Calculate Fisher information and efficiency of your estimator.

e) Is it sufficient estimator of λ ?

f) Test hypotheses H0: λ0=1 against alternative H1 : λ1=1.2 at the 0.025 significance level. Find approximately critical region for n=100.

g) Find power function of the test for n=100 and calculate it approximately. Is it most powerful test?

h) Calculate approximate p-value of the test if total number of calls during first 100 hours occurs S100 =130.

Answer 1