Question

The gamma distribution with shape $ u > 0$ and rate $lambda > 0$ has p.d.f. $frac{lambda^{ u}}{Gamma( u)} x^{ u - 1} e^{-lambda x}$ for $x > 0$. (a) Show that the gamma distribution is an exponential family. (b) Obtain the canonical link and the variance function.

          The gamma distribution with shape $
u > 0$ and rate $lambda > 0$ has p.d.f.
$frac{lambda^{
u}}{Gamma(
u)} x^{
u - 1} e^{-lambda x}$
for $x > 0$.
(a) Show that the gamma distribution is an exponential family.
(b) Obtain the canonical link and the variance function.

$The gamma distribution with shape u > 0 and rate lambda > 0 has p.d.f. fraclambda^ uGamma( u) x^ u - 1 e^-lambda x for x > 0. (a) Show that the gamma distribution is an exponential family. (b) Obtain the canonical link and the variance function.$

Added by Victoria L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Jerelyn Nevil

Step 1

The pdf of the gamma distribution is given by: f(x; v, A) = \frac{A^v x^{v-1} e^{-Ax}}{\Gamma(v)} Now, let's rewrite this in the form of an exponential family distribution: f(x; v, A) = \frac{A^v}{\Gamma(v)} x^{v-1} e^{-Ax} f(x; v, A) = \exp\left(v \log A - Show more…

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The gamma distribution with shape v > 0 and rate A > 0 has p.d.f: f(s;v,A) = (A^v * s^(v-1) * e^(-As)) / Γ(v) for s > 0. (a) Show that the gamma distribution is an exponential family. (b) Obtain the canonical link and the variance function.