1. This question is about ML estimates of transformations of the parameter.
(a) Suppose that X iid
∼ Exponential(\lambda ) where \theta = R++ (i.e. the strictly positive
real numbers). Consider a transformed parameter \tau =( 1)/(1 + \lambda ). Write the
likelihood function in terms of \tau , then maximize it to find the ML estimate
ˆ\tau . Confirm that ˆ\tau =( 1)/(1 + ˆ\lambda ), where ˆ\lambda is the ML estimate of \lambda .
(b) If X ∼ Poisson(\lambda ) where \theta = R++, then P{X = 0; \lambda } = exp(-\lambda ). Explain
why the ML estimate of this probability is exp(-ˆ\lambda ), where ˆ\lambda is the ML
estimate of \lambda