Suppose we observe a Poisson process for a length of time t.

The number of events is X ∼ Poisson(λt). We can show that the maximum likelihood estimator of λ is

λb=

X t

.

Say whether each of the following statements is true or false.

Show your working or reasoning.

a. E(X) = λt and Var(X) = λt.

b. E(λb) = λ and Var(λb) =

λ

t

.c. λbis unbiased, and its variance gets smaller if we observe the process for a longer time interval, t.

d. E(X) = λ and Var(X) = λ.