Gamma Distribution

Modeling waiting times and the sum of exponential variables.

The "Waiting Time" Distribution

The Gamma distribution is a versatile, two-parameter continuous probability distribution that is strictly positive. It's often used to model the waiting time until a specified number of events occur.

Think of it this way: if the time until the *next* bus arrives is modeled by an Exponential distribution, then the time until the *third* bus arrives is modeled by a Gamma distribution. In finance, it can be used to model the size of insurance claims, loan defaults, or operational losses, where the values are always positive and often skewed.

The Formula
The probability density function (PDF) is given by:
f(x;α,β)=βαΓ(α)xα−1e−βxf(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}
  • x>0x > 0.
  • α\alpha (alpha) is the shape parameter.
  • β\beta (beta) is the rate parameter.
  • Γ(α)\Gamma(\alpha) is the Gamma function.
Interactive Gamma Distribution
Adjust the shape (α) and rate (β) parameters to see how the form of the distribution changes. Notice how for large α, it starts to resemble a normal distribution.