Discussion
Suppose \(f(x, \theta)\) is a probability density function of X for which differentiation under integral sign is permissible.
Then \(E\left(\dfrac{d}{d\theta}\log f(X, \theta)\right)\) is :
Answer: D
The expectation is equal to zero.
Differentiate the identity \(\int f(x, \theta)\,dx = 1\) under the integral sign:
\[ \begin{aligned} 0 &= \int \frac{\partial f}{\partial\theta}\,dx = \int \frac{\partial \log f}{\partial\theta}f\,dx \\ &= E\left(\frac{d}{d\theta}\log f(X, \theta)\right) \end{aligned} \]
So the score has mean 0. The Fisher information is the variance of the score, \(E[(d\log f/d\theta)^{2}]\), not its mean. 'Less than zero' and 'less than 1' are not the exact value.
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