How gamma are members of the exponential family prove?

Answer 1

Think you've got this backwards. The exponential probability distribution is a gamma probability distribution only when the first parameter, k is set to 1. Consistent with the link below, if random variable X is distributed gamma(k,theta), then for gamma(1, theta), the random variable is distributed exponentially.

The gamma function in the denominator is equal to 1 when k=1. The denominator will reduce to theta when k = 1. The first term will be X0 = 1.

using t to represent theta, we have

f(x,t) = 1/t*exp(-x/t)

or we can substitute L = 1/t, and write an equivalent function:

f(x;L) = L*exp(-L*x) for x > 0

See:

http://en.wikipedia.org/wiki/Gamma_distribution

[edit]

To the untrained eye the question might seem backwards after a quick Google search, yet qouting wikipedia lacks deeper insight in to the question: What the question is referring to is a class of functions that factor into the following form:

f(y;theta) = s(y)t(theta)exp[a(y)b(theta)] = exp[a(y)b(theta) + c(theta) + d(y)]

where a(y), d(y) are functions only reliant on y and where b(theta) and c(theta) are answers only reliant on theta, an unkown parameter.

if a(y) = y, the distribution is said to be in "canonical form" and b(theta) is often called the "natural parameter"

So taking the gamma density function, where alpha is a known shape parameter and the parameter of interest is beta, the scale parameter. The density function follows as:

f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha)

where gamma(alpha) is defined as (alpha - 1)!

Hence the gamma-density can be factored as follows:

f(y;beta) = {(beta^alpha)*[y^(alpha - 1)]*exp[-y*beta]}/gamma(alpha)

=exp[alpha*log(beta) + (alpha-1)*log(y) - y*beta - log[gamma(alpha)]

from the above expression, the canonical form follows if:

a(y) = y

b(theta) = -beta

c(theta) = alpha*log(beta)

d(y) = (alpha - 1)*log(y) - log[gamma(alpha)]

which is sufficient to prove that gamma distributions are part of the exponential family.