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3. Multivariate Random Variables 173

the one-parameter exponential family defined by (3.8.1). {Hint: Use ideas similar
to those from the Example 3.8.5.}
3.8.5 Consider two random variables X , X whose joint pdf is given by
1 2

with θ(> 0) unknown.
(i) Are X , X independent?
1 2
(ii) Does this pdf belong to the one-parameter exponential family?
3.8.6 Does the multinomial pmf with unknown parameters p , ..., p de-
1
k
fined in (3.2.8) belong to the multi-parameter exponential family? Is the num-
ber of parameters k or k – 1?
3.8.7 Express the bivariate normal pdf defined in (3.6.1) in the form of an
appropriate member of the one-parameter or multi-parameter exponential family
in the following situations when the pdf involves
(i) all the parameters µ , µ , σ , σ , ρ;
1 2 1 2
(ii) µ = µ = 0, and the parameters σ , σ , ρ;
1 2 1 2
(iii) σ = σ = 1, and the parameters µ , µ , ρ;
1 2 1 2
(iv) σ = σ = σ, and the parameters µ , µ , σ, ρ;
1 2 1 2
(v) µ = µ = 0, σ = σ = 1, and the parameter ρ.
1 2 1 2
3.8.8 Consider the Laplace or the double exponential pdf defined as

for –∞ < x, θ < ∞ where θ is referred to as a parameter. Show that f(x; θ)
does not belong to the one-parameter exponential family.
3.8.9 Suppose that a random variable X has the Rayleigh distribution with

where θ(> 0) is referred to as a parameter. Show that f(x; θ) belongs to the
one-parameter exponential family.
3.8.10 Suppose that a random variable X has the Weibull distribution with

where α(> 0) is referred to as a parameter, while β(> 0) is assumed known.
Show that f(x; α) belongs to the one-parameter exponential family.
3.9.1 (Example 3.9.4 Continued) Prove the claim made in the Example
3.9.4 which stated the following: Suppose that X is a random variable
whose mgf M (t) is finite for some t ∈ Τ ⊆ (–∞, 0). Then, it follows
X