Maximum likelihood estimates possess the property of functional invariance, which means that if…

Maximum likelihood estimates possess the property of functional invariance, which means that if &#120579; ̂ is the MLE of &#120579;, and h(&#120579;) is any function of &#120579;, then h(&#120579; ̂) is the MLE of h(&#120579;).

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(&#120582;), then P(X = 0) = e−&#120582;. 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(&#120582;) distribution.

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