Discussion
Which of the following distributions belong to the one parameter exponential family ?
Answer: D
The Laplace distribution with location 0 and scale \(\theta\) belongs to the one-parameter exponential family, so the answer is option D.
Its density is \(\frac{1}{2\theta}e^{-|x|/\theta}\), which can be written as \(h(x)\exp\{\eta(\theta)T(x) - A(\theta)\}\) with \(T(x) = |x|\), \(\eta(\theta) = -1/\theta\) and \(A(\theta) = \ln 2\theta\), and its support does not depend on \(\theta\).
The uniform \(U(0, \theta)\) fails because its support depends on \(\theta\). \(N(\theta, \theta^{2})\) has one parameter but two natural statistics (x and \(x^{2}\)), a curved exponential family and not a one-parameter one. The Gamma option is the one that comes closest: with the scale known it has the form \(\exp\{(\lambda - 1)\ln x - \ldots\}\), but the shape is restricted to the integers 2, 3, 4, ..., and the standard definition of a one-parameter family asks for a parameter space that contains an interval. On a looser definition that allows any parameter set the Gamma option would also qualify; the key chooses D.
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