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Exponential family

For fixed positive integer $r$, the failures-before-the-$r$th-success distribution is a one-parameter exponential family with
$$
\theta=\log(1-p)<0,
\quad
T(x)=x,
\quad
A(\theta)=-r\log(1-e^\theta),
$$
and carrier $h(x)=\binom{x+r-1}{x}$.

Negative binomial exponential family

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 206 / 1 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 5J / Solution

Negative binomial exponential family

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Negative binomial exponential family

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