Discussion
Which of the following probability laws is not a member of an exponential family ?
Answer: C
The law that is not a member of the (one-parameter) exponential family is N(θ, θ2).
Writing the density,
\[ \begin{aligned} f(x) &\propto \frac{1}{\theta}\exp(Q), \\ Q &= -\frac{x^{2}}{2\theta^{2}} + \frac{x}{\theta} - \frac{1}{2} \end{aligned} \]
the exponent has two terms, \(x^{2}\) and x, with coefficients \(-1/(2\theta^{2})\) and \(1/\theta\) depending on the single parameter. It cannot be written as \(c(\theta)T(x)\) with one statistic, so it is a curved family, not a one-parameter exponential family.
The others are members: \(\theta(1 - x)^{\theta - 1} = \exp[(\theta - 1)\log(1 - x) + \log\theta]\); the Bernoulli is \(\exp[x\log\frac{\theta}{1 - \theta} + \log(1 - \theta)]\); and for \(N(\theta, \theta)\) the x term reduces to the constant 1, leaving only \(-x^{2}/(2\theta)\).
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