17.3 Gamma Distribution: Probability Density Function (PDF)

The Gamma Distribution is a continuous probability distribution characterized by two parameters, frequently employed to model phenomena such as waiting times, durations of life, and reliability events.

Parameters

The Gamma Distribution is defined by the following parameters:

• Shape parameter: $\alpha$ (also denoted as $k$)
• Rate parameter: $\lambda$ (where $\lambda = 1 / \theta$, and $\theta$ is the scale parameter, also denoted as $\beta$)

Probability Density Function (PDF) Formula

For a random variable $X$ that follows a Gamma Distribution, its probability density function (PDF) is given by:

$$f(x; \alpha, \lambda) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}, \quad \text{for } x > 0$$

Where:

• $x$: The random variable ($x > 0$).
• $\alpha$: The shape parameter ($\alpha > 0$).
• $\lambda$: The rate parameter ($\lambda > 0$).
• $\Gamma(\alpha)$: The Gamma function evaluated at $\alpha$.

The Gamma Function, $\Gamma(\alpha)$

The Gamma function is a generalization of the factorial function to complex and real numbers. For a positive real number $\alpha$, it is defined as:

$$\Gamma(\alpha) = \int_0^\infty t^{\alpha-1} e^{-t} dt$$

For positive integers $n$, the Gamma function simplifies to $\Gamma(n) = (n-1)!$.

Key Characteristics

• Skewness: The distribution exhibits right-skewness when the shape parameter $\alpha < 1$.
• Symmetry: As the shape parameter $\alpha$ increases, the distribution becomes more symmetric.
• Special Cases:
  • When $\alpha = 1$, the Gamma Distribution reduces to the Exponential Distribution with rate parameter $\lambda$.
  • When $\alpha = \nu/2$ and $\lambda = 1/2$, the Gamma Distribution is equivalent to the Chi-Square Distribution with $\nu$ degrees of freedom.

Summary

• PDF: $f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\lambda x}$, for $x > 0$
• Shape Parameter: $\alpha$
• Rate Parameter: $\lambda$
• Gamma Function: $\Gamma(\alpha)$

Applications

The Gamma distribution is widely used in various fields to model:

• Waiting times: For events that occur at a constant average rate, such as the time until the $k$-th customer arrives at a store.
• Reliability engineering: To model the lifetime of components or systems.
• Queuing theory: To describe arrival or service times.
• Finance: In modeling insurance claims or the distribution of investment returns.

Interview Questions

• What is the Gamma distribution and when is it typically used?
• Define the PDF of the Gamma distribution and explain the role and interpretation of its parameters ($\alpha$ and $\lambda$).
• What is the significance of the Gamma function ($\Gamma(\alpha)$) in the context of this distribution?
• How does the shape parameter $\alpha$ influence the overall shape and behavior of the Gamma distribution?
• Clarify the relationship between the rate parameter ($\lambda$) and the scale parameter ($\theta$ or $\beta$).
• Explain the connection between the Gamma distribution and the Exponential distribution.
• Describe how the Gamma distribution relates to the Chi-Square distribution.
• Provide examples of real-world scenarios where the Gamma distribution is a suitable model.
• What happens to the shape of the Gamma distribution as the shape parameter $\alpha$ increases?
• How would you conceptually generate a random variable following a Gamma distribution in a programming context (e.g., Python)?