Gamma Distribution definition, formula and applications

In probabilistic statistics, the gamma distribution is a two-parameter family of continuous probability distributions which is widely used in different sectors. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of it.

Gamma distribution is a crucial concept in probability theory and statistics, widely used in various scientific fields such as finance, engineering, and machine learning. In this article, we will explore the fundamentals of the gamma distribution, its properties, applications, and uses in real-world scenarios.

What is the Gamma Distribution?

At its core, the Gamma distribution is a continuous probability distribution that models the waiting time until the kth event in a Poisson process. A Poisson process is a random process that describes the number of events occurring in a fixed interval of time or space, with events happening independently and at a constant average rate (denoted by λ).

A continuous random variable X is said to have a generalized gamma distribution with parameter α and β if its probability density function is defined as,

Where α, β are the two parameters and Γ(α) is the gamma function, which generalizes the factorial function for real and complex numbers. Also, α,β>0. Gamma distribution is a continuous probability distribution.

The gamma distribution is a two-parameter family of continuous probability distributions. It is often used to model waiting times, reliability analysis, and queuing systems. The parameters are:

1. Shape parameter (α): Determines the shape of the distribution.
2. Scale parameter (β): Stretches or shrinks the distribution along the x-axis.

Example: Modeling the Time Until the 3rd Customer Arrives

Imagine you’re running a small coffee shop. You observe that customers arrive at an average rate of 5 per hour (λ = 5). You want to estimate the probability that it takes longer than 30 minutes (0.5 hours) for the third customer to arrive. In this case, you can use the Gamma distribution with α = 3 (since you’re interested in the third customer) and λ = 5.

To find the probability P(X > 0.5), you would need to calculate the integral of the Gamma PDF from 0.5 to infinity. Researchers often use statistical software or look up values in a Gamma distribution table to perform this task.

Origin/History

According to Johnson et al. (1994, p. 343), the genesis of the gamma distribution may be traced back to Laplace (1836), who defined it as the distribution of a “precision constant.” Waiting times were modeled using the gamma distribution. In life testing, for example, the time it takes to reach “death” is a random variable with a gamma distribution (Hogg et al. 2013, p. 156). In Bayesian statistics, the Gamma distribution serves as a conjugate prior for many scale parameters, such as the parameter in an exponential distribution or a normal distribution with a known mean.

When to Use Gamma Distribution

If you have a sharp lower bound of zero but no sharp upper bound, a single mode, and a positive skew, use the Gamma distribution with α > 1. In this situation, the Log Normal distribution is also an option. Gamma is particularly useful for encoding arrival times for groups of events. When you want to employ a bell-shaped curve for a positive-only quantity, a gamma distribution with a large value for α is also suitable.

Properties of Gamma Distribution

There are some important properties as following,

• It thas two parameter  α and β.
• Mean of variate is α/β.
• Variance of variate  is α/ β².
• Characteristic function  is .
• Moment generating function  is .
• The measure of skewness β₁=.
• Measure of kurtosis, β₂= . These measures show that gamma distribution is positively skewed and leptokurtic.
• If the value of  β=1; mean= variance, if  β>1; mean<variance and if  β<1; mean>variance.

Applications of the Gamma Distribution

The Gamma distribution finds applications in a wide array of fields due to its flexibility in modeling continuous, positive data:

• Finance: Modeling insurance claim sizes, credit risk, and option pricing. The Gamma distribution’s skewed nature effectively represents events where large values are less frequent but have a significant impact.
• Engineering: Reliability analysis, modeling the time to failure of components or systems, and in queuing theory, analyzing waiting times in service systems.
• Meteorology: Modeling rainfall amounts, wind speeds, and other meteorological variables.
• Healthcare: Analyzing survival times of patients, modeling the incubation period of diseases, and studying the distribution of medical costs.
• Telecommunications: Modeling the duration of phone calls, data transmission rates, and network traffic.
• Actuarial Science: Estimating insurance losses and projecting future liabilities.
• Bayesian Statistics: Serving as a prior distribution for variance parameters in Bayesian models.

Relations to Others Distribution

There are some relations to other distribution-

• The Gamma distribution is the continuous analog of the Negative Binomial distribution.
• The special case where α=1 is an Exponential distribution.
• The special case where α=ν/2 and β=1/2 is a chi-square parametrized by ν.

The Gamma distribution is intimately related to several other important probability distributions:

• Exponential Distribution: As mentioned earlier, when the shape parameter α = 1, the Gamma distribution reduces to the Exponential distribution. The Exponential distribution models the waiting time until the first event in a Poisson process, while the Gamma distribution generalizes this to the kth event.
• Erlang Distribution: The Erlang distribution is a special case of the Gamma distribution where the shape parameter α is a positive integer. It specifically models the waiting time until the kth event in a Poisson process. The Erlang distribution is commonly used in queuing theory and telecommunications.
• Chi-Squared Distribution: The Chi-squared distribution is a special case of the Gamma distribution where α = ν/2 (where ν is the degrees of freedom) and λ = 1/2. Researchers and analysts widely use the Chi-squared distribution in hypothesis testing and confidence interval estimation, particularly for analyzing categorical data and testing the goodness-of-fit of statistical models.
• Normal Distribution: As the shape parameter α becomes large (approaches infinity), the Gamma distribution approaches a Normal distribution. This is due to the Central Limit Theorem, which states that the sum of a large number of independent and identically distributed random variables tends to follow a Normal distribution.

Challenges and Considerations

While the Gamma distribution is a powerful tool, it’s important to be aware of its limitations:

• Data Requirements: Accurately estimating the shape and rate parameters requires a sufficient amount of data. Small sample sizes can lead to inaccurate parameter estimates and unreliable predictions.
• Parameter Estimation: Estimating the parameters of the Gamma distribution can be challenging, especially for complex datasets. Researchers and analysts commonly use maximum likelihood estimation (MLE) and the method of moments for parameter estimation.
• Goodness-of-Fit: Before using the Gamma distribution to model a dataset, it’s essential to assess whether the distribution provides a good fit to the data. Analysts can use goodness-of-fit tests, such as the Kolmogorov-Smirnov test or the Chi-squared test, to evaluate the adequacy of the Gamma distribution.
• Computational Complexity: Calculating probabilities and quantiles for the Gamma distribution can be computationally intensive, particularly for large values of the shape parameter. Statistical software packages typically provide functions for working with the Gamma distribution, but users must recognize the computational challenges.

How to Use Gamma Distribution in Python

Python provides several libraries, such as scipy.stats, to work with the gamma distribution. Below is a simple example:

Conclusion

The Gamma distribution is a versatile and powerful tool for modeling continuous, positive data. Its flexibility stems from its two parameters, shape and rate (or scale), which allow it to capture a wide range of shapes and skewness. Its connections to other important distributions like the Exponential, Erlang, Chi-squared, and even the Normal distribution make it a central component of statistical modeling.

By understanding the Gamma distribution’s properties, parameters, and relationships to other distributions, you can unlock its full potential and apply it effectively to a wide range of real-world problems. While its mathematical foundation can seem daunting at first, the effort spent understanding this distribution is well worth the investment, providing you with a valuable tool for analyzing and interpreting data in various fields. From finance and engineering to healthcare and telecommunications, the Gamma distribution offers a powerful lens through which to understand the random processes that shape our world.