Question 11.11: The N(θ, σ²) (σ known) p.d.f. is of the exponential type....

In fact,

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

and this is of the form(4)

with C(\theta)={\frac{1}{\sqrt{2\pi}\sigma}}e^{-{\frac{\theta^{2}}{2\sigma^{2}}}},\,Q(\theta)={\frac{\theta}{\sigma^{2}}} strictly increasing, T(x)=x,\;\mathrm{and}\;h(x)=e^{-x^{2}/2\sigma^{2}}.