Final answer:
The question is seeking the lower bound on the variance for an unbiased estimator of a discrete random variable, indicating an application of the Cramér-Rao lower bound. The PMF provided resembles a Poisson distribution, and the Cramér-Rao lower bound would be obtained from the inverse of the Fisher Information of the PMF.
Step-by-step explanation:
The question is asking for the lower bound on the variance of any unbiased estimate, a(x), for a discrete random variable x. This is likely referring to the Cramér-Rao lower bound, which in statistics gives a theoretical lower bound on the variance of unbiased estimators.
Given the probability mass function (PMF) provided, which resembles the Poisson distribution due to the factor of e raised to the power -A, the lower bound would be dependent on the inverse of the Fisher Information in the PMF.
To calculate the Fisher Information, we would need to derive the score function and calculate its expected value squared, which is outside the scope of this summary.
Therefore, it is important to note that while the question initially appears to discuss uniform distribution, it is actually concerned with a Poisson-like PMF.
The information provided about uniform distribution is not directly relevant to the computation of the lower bound of a discrete random variable's variance.
Your complete question is: Problem 3: We observe a value of the discrete random variable x. Pr(x=i∣A)=
i!
A
i
e
−A
,i=0,1,2,…, where A is nonrandom. (a) What is the lower bound on the variance of any unbiased estimate,
a
^
(x) ? (b) Assuming n independent observations, find an
a
^
(x) that is efficient.