4.23 Let X1, . . . , Xn be i.i.d. observations with the pdf or pmf f (x|θ), where θ is a univariate parameter. Here, the pdf is with respect to the Lebesgue measure, whereas the pmf may be regarded as a pdf with respect to the countingmeasure

[see below (4.36)]. Obtain the Fisher information (4.75) for the following cases:

(i) X1 ∼ Bernoulli(θ), so that f (x|θ) = θx (1 − θ)1−x, x= 0, 1, where θ ∈ (0, 1).

(ii) X1 ∼ Poisson(θ), so that f (x|θ) = e

−θ θx x!, x= 0, 1, . . . , where θ > 0.

(iii) X1 ∼ Exponential(θ), so that f (x|θ) = 1

θ

e

−x/θ, x≥ 0, where θ > 0.

(iv) X1 ∼ N(θ, θ2), so that f (x|θ) = 1 √
2πθ2 exp −(x − θ)2 2θ2 , −∞ < x < ∞, where θ ∈ (−∞,∞).