The random variables 1 2 , ,..., n X X X are independent from each other and all follow a poison distribution with param

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The random variables 1 2 , ,..., n X X X are independent from
each other and all follow a poison distribution with parameter
£ . i. Derive the maximum likelihood estimator of £ based
on 1 2 , ,..., n X X X . You are not required to verify that your
answer corresponds to a maximum. [4] ii. Derive an expression for
an approximate 95% confidence interval for £ under the
situation in part (i) using the Cramer – Rao lower bound. [4]
Suppose that instead of observing the values of 1 2 , ,..., n X X X
precisely, we only observe that for K of these variables we have 0
i X £ , while for the remaining variables we have 0 i X
£ iii. Derive the maximum likelihood estimator of £ when
only this information is available. You are not required to verify
that your answer corresponds to a maximum.