- #1
Mogarrr
- 120
- 6
Homework Statement
For the negative binomial distribution, with r known, describe the natural parameter space
Homework Equations
the pmf for the negative binomial distribution with parameters r and p can be
1) [itex]P(X=x|r,p)= \binom {x-1}{r-1}p^{r}(1-p)^{x-r} [/itex] where [itex]x=r,r+1,... [/itex], or
2) [itex]P(Y=y|r,p)= \binom {y+r-1}{y}p^{r}(1-p)^{y} [/itex] where [itex]y=0,1,... [/itex].
A distribution, like the one above where r is known, is a member of the exponential family of distributions. An exponential distribution is one that can be expressed as...
[itex]h(x)c^{*}(\eta) exp(\sum_{i=1}^{k} \eta_i t_i(x)) [/itex]
The parameter space are the values of [itex]\eta [/itex] such that [itex]\sum_A h(x) exp(\sum_{i=1}^{k} \eta_i t_i(x)) < \infty [/itex] where [itex]A [/itex] is the support of the pmf.
The Attempt at a Solution
Rewriting the 2nd pmf for the negative binomial distribution, as an exponential distribution, I have
[itex]h(y) = \binom {y+r-1}{y} [/itex], [itex]c(p) = p^{r} \cdot I_(0,1)(p) [/itex], [itex]t_1(y)=y [/itex], and [itex]w_1(p) = ln(1-p) [/itex].
Then I let [itex]\eta = w_1(p) [/itex], and find the values for [itex]\eta [/itex] where the sum converges.
I have [itex]\sum_{y=0}^{\infty} \binom{r+y-1}{y}(e^{\eta})^{y} [/itex], and I don't recognize this sum as anything that converges.
Any help would be appreciated.