When to Use the Gamma Distribution

The gamma distribution models waiting times for events that occur at a constant average rate. By adjusting the shape parameter $k$ and scale parameter $\mathrm{\backslash theta}$, it describes a range of phenomena from the lifetime of electronic components to rainfall amounts. In essence, the gamma family generalizes the exponential distribution beyond the memoryless case.

Probability Density

The gamma density function is expressed as

$f\left(x\right)=\frac{x^{k-1}}{e^{-\frac{x}{\mathrm{\backslash theta}}}}{\mathrm{\backslash theta}}^k\mathrm{\backslash Gamma}\left(k\right)$

where $\mathrm{\backslash Gamma}$ is the gamma function. This density begins at zero, rises to a peak, and then decays exponentially.

Cumulative Distribution

The CDF involves the lower incomplete gamma function. This calculator approximates it by summing a series until convergence. The resulting probability gives the likelihood that a gamma-distributed variable is less than or equal to $x$.

Why This Matters

By tuning $k$ and $\mathrm{\backslash theta}$, the gamma distribution handles data that exhibit skewness and is widely used in queueing theory, climatology, and reliability engineering. It is also a building block for other distributions such as chi-squared.