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3. Multivariate Random Variables 143
Example 3.8.4 Consider a random variable X having a negative exponen-
tial pdf, defined in (1.7.36). The pdf is θ exp{(x θ)/θ}I(x > θ) with θ > 0,
1
where I(.) is an indicator function. Recall that I(A) is 1 or 0 according as the
set A or A is observed. The term I(x > θ) can not be absorbed in the expres-
c
sions for a(θ), b(θ), g(x) or R(x), and hence this distribution does not belong
to a one-parameter exponential family. !
Example 3.8.5 Suppose that a random variable X has the uniform distribu-
tion, defined in (1.7.12), on the interval (0, θ) with θ > 0. The pdf can be
rewritten as f(x; θ) = θ I(0 < x < θ). Again, the term I(x > θ) can not be
1
absorbed in the expressions for a(θ), b(θ), g(x) or R(x), and hence this distri-
bution does not belong to a one-parameter exponential family. !
