For problems that are related to gamma-family probability distributions.

A random variable $X$ that is gamma-distributed with shape $k$ and scale $\theta$ is denoted by $$X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta)$$

The probability density function using the shape-scale parametrization is $$f(x;k,\theta) = \frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k\Gamma(k)} \quad \text{ for } x > 0 \text{ and } k, \theta > 0$$

Here $\Gamma(k)$ is the gamma function evaluated at $k$.

The cumulative distribution function is the regularized gamma function: $$F(x;k,\theta) = \int_0^x f(u;k,\theta)\,du = \frac{\gamma\left(k, \frac{x}{\theta}\right)}{\Gamma(k)}$$

where $\gamma(k, x/ \theta)$ is the lower incomplete gamma function.

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