LR Test for Exponential Family of Distributions: The exponential family of distributions, both discrete and continuous, based on a parameter θ is defined by:
f (x |theta) = c(x)d(theta)exp[a(theta)b(x)]
Show that N(μ,1) is a special case of this family. That is, determine the values for the functions a, b, c, and d.
Hint: The density of $N(\mu, 1)$ is $$f(x|\mu) = \frac 1{\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2}} $$
Just expand the square.
details: $$f(x|\mu) = \frac 1{\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2}} = \frac 1{\sqrt{2\pi}} e^{-x^2/2} e^{-\mu^2/2} e^{x\mu} $$ so you find (this is not the lonly solution):
$$ c(x) = \frac 1{\sqrt{2\pi}} e^{-x^2/2} \\ d(\mu) = e^{-\mu^2/2} \\ a(\mu) = \mu \\ b(x) = x $$