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The family of beta distributions is a conjugate family of prior distributions for samples from a negative binomial distribution with a known value of the parameterrand an unknown value of the parameterp(0<p <1).

Consider a single observation x from a negative binomial distribution with parameters r and p, where the value of r is known and the value of p is unknown. Then the p.f of X has the form

\(f\left( {x|p} \right) \propto {p^r}{q^x}\).

If the prior distribution of p is a beta distribution with parameters\(\alpha \,\,and\,\,\beta \), then the prior p.d.f.\(\xi \left( p \right)\)has the form

\(\xi \left( p \right) \propto {p^{\alpha - 1}}{q^{\beta - 1}}\)

Therefore, the posterior p.d.f\(\xi \left( {p|x} \right)\)has the form

\(\begin{aligned}\xi \left( {p|x} \right) \propto \xi \left( p \right)f\left( {p|x} \right)\\ \propto {p^{\alpha + r - 1}}{q^{\beta + x - 1}}\end{aligned}\)

This expression can be recognized as being, except for a constant factor, the p.d.f of beta distribution with parameters\(\alpha + r\,\,\,and\,\,\,\beta + x\). Since this distribution will be the prior distribution of p for future observations, it follows that the posterior distribution after any number of observations will also be the beta distribution.