Question 11.12: The N(μ, θ) (μ known) p.d.f. is of the exponential type....

Here

$f(x;\theta)={\frac{1}{{\sqrt{2\pi\theta}}}}e^{-{\frac{1}{2\theta}}(x-\mu)^{2}},$

and this is of the form (4)

with $C(\theta)={\frac{1}{\sqrt{2\pi\theta}}},\ Q(\theta)=-{\frac{1}{2\theta}}$ strictly increasing (since ${\frac{d}{d\theta}}(-{\frac{1}{2\theta}})={\frac{1}{2\theta^{2}}}\gt 0),\;T(x)=(x-\mu)^2,\mathrm{and}\;h(x)=1.$