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Proof that $ I(θ) = E[(s(θ;X))^2] $

Recall that:

$ \begin{align} \bar Var(Y) &= E[(Y-E(Y))^2]\\ &= E[Y^2-2YE[Y]+(E[Y])^2]\\ &= \frac{\mu_0}{2 \pi a \cdot b} \end{align} $

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