We recommend reading this documentation on pkgdown which renders math nicely.
https://pkg.mitchelloharawild.com/distributional/reference/dist_tweedie.html
In the following, let \(X\) be a Tweedie random variable with
parameters mean = \(\mu > 0\), dispersion = \(\phi > 0\), and
power = \(p \in (1, 2)\).
Support: \(\{0\} \cup (0, \infty)\)
Mean: \(\mu\)
Variance: \(\phi \mu^p\)
Probability density function (p.d.f):
The Tweedie distribution does not have a closed-form density. Instead,
it is defined as a Poisson sum of Gamma random variables, with
\(P(X = 0) = \exp(-\lambda)\) and, for \(x > 0\),
$$
f(x) = \sum_{j=1}^{\infty} \frac{\lambda^j e^{-\lambda}}{j!}
\frac{x^{j \alpha - 1} e^{-x / \gamma}}{\gamma^{j \alpha} \Gamma(j \alpha)}
$$
where \(\lambda = \mu^{2 - p} / (\phi (2 - p))\) is the Poisson rate,
\(\alpha = (2 - p) / (p - 1)\) is the Gamma shape per event, and
\(\gamma = \phi (p - 1) \mu^{p - 1}\) is the Gamma scale. The density
is evaluated using the series expansion of Dunn & Smyth (2005).
Cumulative distribution function (c.d.f):
The cumulative distribution function is evaluated numerically from the
density series, as it does not have a closed-form expression.