Let X1,...,Xn ∼ iid Pois(θ), and we wish to estimate φ = Pθ(X = 0) = e−θ. (a)  Find the MLE φˆ. (b)  Use Jensen’s inequality to derive a result comparing E(φˆ) to φ. In fact, since g(t) = e−t is not a linear function, the inequality this result is strictly, implying that φˆ is a biased estimator. (c)  Consider Yi = I (Xi = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution

Let X₁,...,X_n ∼ iid Pois(θ), and we wish to estimate φ = P_θ(X = 0) = e^−θ.

1. (a)  Find the MLE φˆ.

2. (b)  Use Jensen’s inequality to derive a result comparing E(φˆ) to φ. In fact, since g(t) = e^−t is not

   a linear function, the inequality this result is strictly, implying that φˆ is a biased estimator.

3. (c)  Consider Y_i = I (X_i = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution