The third question in the attachment is:

I assume that the X's are independent ....?Suppose that form a random sample from a Poisson distribution with unknown mean , and let , determine the value of a constant such that the estimator is an unbiased estimator of .

Distribution of :

It's well known and easy to prove that the sum of independent Poisson random variables with parameters is a Poisson random variable with parameter .

Therefore follows a Poisson distribution with parameter .

Unbiased estimator:

You require .

Calculation of :

The calculation follows exactly the same steps as the well known calculation for the moment generating function of a Poisson random variable.

It is well known that if follows a Poisson distribution with parameter then the moment generating function of is .

Substituting and you get:

.

It should be clear how to go from here.