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 .
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.