Let X be a Poisson (θ) random variable. Unbiased estimatorShow that (−1)X is an unbiased estimator for e−2θ This is a fairly bad estimator for a number of reasons - so this exercise helps show why unbiasedness is not the most important criterion for an estimator.

Question

sa3b4or9i9 · Accepted Answer

Let us derive this in detail, to address your confusion.By definition, an estimator&amp;nbsp;μ^&amp;nbsp;is an unbiased estimator for some quantity&amp;nbsp;μ&amp;nbsp;if&amp;nbsp;E[μ^]=μ.So what we need to show is that&amp;nbsp;E[(−1)X]=e−2θ.Writing the definition of expectation,E[(−1)X]=∑n=0∞(−1)nP{X=n}=∑n=0∞(−1)ne−θθnn!=e−θ∑n=0∞(−θ)nn!where we used the fact that&amp;nbsp;X~Poisson(θ)&amp;nbsp;for the second equality.To conclude, we recall the definition of exponential: for any&amp;nbsp;x∈R,ex=∑n=0∞xnn!. Thus,E[(−1)X]=e−θ∑n=0∞(−θ)nn!=e−θe−θ=e−2θ&amp;nbsp;concluding the proof.

Let X be a Poisson (\(\displaystyle\theta\)) random

Answered question

Answer & Explanation

New Questions in College Statistics