Maximum likelihood estimates possess the property of functional invariance, which means that if  is the MLE of θ, and h(θ) is any function of θ, then h() is the MLE of h(θ).
a. Let X ∼ Bin(n, p) where n is known and p is unknown. Find the MLE of the odds ratio p/(1 − p).
b. Use the result of Exercise 5 to find the MLE of the odds ratio p/(1 − p) if X ∼ Geom(p).
c. If X ∼ Poisson(λ), then P(X = 0) = e−λ. Use the result of Exercise 6 to find the MLE of P(X = 0) if X1, . . . , Xn is a random sample from a population with the Poisson(λ) distribution.