From Poisson to Pascal Distribution Through Gamma Mixing

The Poisson distribution does not literally turn into the Pascal distribution. The connection appears when the Poisson rate is not fixed: model that rate as a Gamma-distributed random variable, then integrate it out. The resulting marginal count distribution is the Pascal distribution, also called the negative binomial distribution.

This construction is useful for count data with more variability than a Poisson model permits.

Probability Mass Functions, Not Probability Density Functions

Both the Poisson and Pascal distributions are discrete distributions. Therefore, they have probability mass functions (PMFs), not probability density functions (PDFs).

Let a count variable $X$, conditional on a rate $\lambda$, follow a Poisson distribution:

\[X \mid \lambda \sim \operatorname{Poisson}(\lambda).\]

Its PMF is

\[P(X=k \mid \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}, \qquad k=0,1,2,\ldots\]

The rate $\lambda$ is continuous, so its distribution does have a PDF.

A Random Poisson Rate

Assume the Poisson rate follows a Gamma distribution with shape $r>0$ and rate $\beta>0$:

\[\lambda \sim \operatorname{Gamma}(r, \beta).\]

Using the shape-rate parameterization, the Gamma density is

\[f_{\lambda}(\lambda) = \frac{\beta^r}{\Gamma(r)} \lambda^{r-1} e^{-\beta\lambda}, \qquad \lambda>0.\]

Here, $\lambda$ can vary across observations, time periods, or individuals. This variability is the reason the marginal count can have a larger variance than its mean.

Marginalizing the Rate

To obtain the unconditional distribution of $X$, integrate the conditional Poisson PMF over the Gamma density:

\[P(X=k) = \int_0^\infty P(X=k \mid \lambda) f_{\lambda}(\lambda) \, d\lambda.\]

Substitution gives

\[\begin{aligned} P(X=k) &= \int_0^\infty \frac{\lambda^k e^{-\lambda}}{k!} \cdot \frac{\beta^r}{\Gamma(r)} \lambda^{r-1}e^{-\beta\lambda} \, d\lambda \\ &= \frac{\beta^r}{k!\Gamma(r)} \int_0^\infty \lambda^{k+r-1}e^{-(\beta+1)\lambda} \, d\lambda \\ &= \frac{\Gamma(k+r)}{k!\Gamma(r)} \left(\frac{\beta}{\beta+1}\right)^r \left(\frac{1}{\beta+1}\right)^k. \end{aligned}\]

Set

\[p = \frac{\beta}{\beta+1}, \qquad 1-p = \frac{1}{\beta+1}.\]

Then

\[P(X=k) = \binom{k+r-1}{k} p^r(1-p)^k, \qquad k=0,1,2,\ldots\]

This is the Pascal, or negative binomial, PMF under the convention that $X$ counts failures before the $r$th success, where each trial succeeds with probability $p$.

Why the Mixture Matters

For a Poisson distribution with fixed $\lambda$,

\[\operatorname{E}[X] = \operatorname{Var}(X) = \lambda.\]

After Gamma mixing, the negative binomial distribution has

\[\operatorname{E}[X] = \frac{r(1-p)}{p}, \qquad \operatorname{Var}(X) = \frac{r(1-p)}{p^2}.\]

Because $0<p<1$, the variance is larger than the mean. This property, called overdispersion, makes the Poisson-Gamma mixture a practical model for real count data whose observations are more heterogeneous than a fixed-rate Poisson model can represent.

Summary

\[\boxed{ X \mid \lambda \sim \operatorname{Poisson}(\lambda), \quad \lambda \sim \operatorname{Gamma}(r,\beta) \quad \Longrightarrow \quad X \sim \operatorname{NegativeBinomial}(r,p) }\]

with $p = \beta/(\beta+1)$. The Gamma distribution supplies a continuous PDF for the uncertain rate, while the resulting Poisson mixture remains a discrete distribution with a Pascal/negative-binomial PMF.