6. Consider the pdf
f(x|θ1,θ2) =
( θ2
2πx3
)1/2
exp
[−θ2(x −θ1)2
2θ21x
]
, 0 < x < ∞, θ1 > 0,θ2 > 0.
[6 pts]
(a) Verify that the inverse-Gaussian family is a member of the exponential family. [Identify h(x),
c(θ), wi(θ), and ti(x).]
[6 pts]
(b) Describe the natural parameter space, and write the pdf in terms of the natural parameters.

[6 pts]
(c) Obtain E(X) and E(1/X) where X is the inverse Gaussian random variable. Write your final
solution as a function of θ1 and θ2, as relevant. You are not allowed to obtain these expectations
by integration.