Unbaised estimator of $\ e^{- 2 \lambda } $ is t(x) = $ \ (-1)^x $ Prove that
Unbaised estimator of $\ e^{- 2 \lambda  } $ is t(x) =  $ \ (-1)^x $  ,
when x follows poisson distribution with parameter $\lambda $
Sorry to ask the direct question but I need a starting hint to solve the question , actually last time I studied statistical inference was 1 year back , I would be great if anyone could just give some starting steps .

If you want to show it's unbiased then you need to show it has the correct expectation. Then your hint is that you're going to want to think about the probability generating function of a Poisson. Hopefully that's enough.
