Final answer:
The Negative Binomial distribution can be expressed in the form of an exponential family distribution by identifying the components that correspond to the exponential family definition. With known r and success probability p, the corresponding components such as the base measure h(x), natural parameter η(θ), sufficient statistic T(x), and log-partition function A(θ) can be identified from the probability mass function.
Step-by-step explanation:
To show that the Negative Binomial distribution for a random sample X1, …, Xn is an exponential family distribution, let us first understand the form of an exponential family distribution. A distribution belongs to the exponential family if its probability density function (pdf) or probability mass function (pmf) can be expressed in the form:
f(x| θ) = h(x) exp[η(θ)T(x) - A(θ)]
where h(x) is the base measure, η(θ) is the natural parameter, T(x) is the sufficient statistic, and A(θ) is the log-partition function.
The Negative Binomial distribution with known r and success probability p has the pmf:
P(X = k) = πⁿ (1 - p)^k where k = 0, 1, 2, … and π = [Π r-1ₜ₀ (r+ₜ-1)] / ₜ! (this is the number of ways of arranging r-1 successes in the first k+ₜ trials).
Examining the probability mass function, we can identify the components that correspond to the exponential family as follows:
h(k) = πⁿ / ₜ!
η(p) = log((1-p)/p)
T(k) = k
A(p) = -r log(p)
With these components, we can express the Negative Binomial distribution in the exponential family form, thus confirming that it is an exponential family distribution.