Introduction
Probability distributions are essential tools in data analysis and modeling. Gamma distribution is a popular continuous probability distribution that can be used to model various real-world phenomena such as waiting times, incomes, and insurance claims. In this article, we will explore the intuition, derivation, and examples of the gamma distribution.
What is Gamma Distribution?
Gamma distribution is a continuous probability distribution that has two parameters, shape (α) and scale (β). It is often denoted as Gamma(α,β). The probability density function (PDF) of gamma distribution is given as:
where x ≥ 0, α > 0, β > 0, and Γ(α) is the gamma function.
The gamma distribution has a variety of shapes depending on the values of its parameters. When α = 1, the gamma distribution reduces to the exponential distribution. When α is an integer, the gamma distribution reduces to the Erlang distribution.
Derivation of Gamma Distribution
The gamma distribution can be derived in various ways. One way is to use the moment generating function (MGF) of a random variable X. The MGF of X is defined as:
Using the MGF, we can derive the probability density function (PDF) of X. For the gamma distribution, the MGF is given as:
Taking the derivative of the MGF and simplifying, we obtain the PDF of the gamma distribution as shown earlier.
Properties of Gamma Distribution
Gamma distribution has several important properties. Some of these properties include:
- Mean: E[X] = αβ
- Variance: Var(X) = αβ^2
- Mode: (α-1)β
- Skewness: Skewness(X) = 2/√α
- Kurtosis: Kurtosis(X) = 6/α + 3
Types of Gamma Distribution
There are two common types of gamma distribution, namely Erlang distribution and Chi-square distribution.
Erlang Distribution
The Erlang distribution is a special case of the gamma distribution, where the shape parameter (α) is an integer. The Erlang distribution is often used to model the waiting times in queuing systems.
Chi-Square Distribution
The Chi-square distribution is also a special case of the gamma distribution, where the shape parameter (α) is an integer and the scale parameter (β) is 2. The Chi-square distribution is often used in hypothesis testing and goodness-of-fit tests.
Applications of Gamma Distribution
Gamma distribution has several applications in various fields such as finance, engineering, physics, and social sciences. Some of the common applications of gamma distribution are:
- Waiting Time: Gamma distribution can be used to model the waiting times in queuing systems such as waiting times in a hospital emergency department or waiting times at a call center.
- Income Distribution: Gamma distribution can be used to model the income distribution of a population. It has been found that the income distribution of most countries follows a gamma distribution.
- Insurance Claims: Gamma distribution can be used to model the frequency and severity of insurance claims. It can help insurance companies to estimate the risk and set premiums.
Examples of Gamma Distribution
Let’s look at some examples to understand the practical applications of gamma distribution.
Waiting Time
Suppose we want to model the waiting time of customers at a restaurant. We can assume that the waiting time follows a gamma distribution with shape parameter α = 3 and scale parameter β = 5. The probability density function of the waiting time is given as:
where x ≥ 0.
Income Distribution
Suppose we want to model the income distribution of a country. We can assume that the income follows a gamma distribution with shape parameter α = 2 and scale parameter β = 10000. The probability density function of the income is given as:
where x ≥ 0.
Insurance Claims
Suppose we want to model the number of insurance claims in a year. We can assume that the number of claims follows a gamma distribution with shape parameter α = 5 and scale parameter β = 1000. The probability mass function of the number of claims is given as:
where k is a non-negative integer.
Advantages and Disadvantages of Gamma Distribution
Gamma distribution has several advantages and disadvantages. Some of the advantages include:
- Flexibility: Gamma distribution is a flexible distribution that can take on a variety of shapes depending on its parameters.
- Goodness of Fit: Gamma distribution can often provide a good fit to real-world data.
- Computational Ease: Gamma distribution has a simple and well-known probability density function that can be easily computed.
Some of the disadvantages of gamma distribution include:
- Limited Domain: Gamma distribution is only defined for non-negative values of X.
- Limited Skewness: Gamma distribution has limited skewness, which can be a disadvantage in certain applications.
- Complex Parameter Estimation: Estimating the parameters of gamma distribution can be complex and time-consuming in certain cases.
Conclusion
In this article, we explored the intuition, derivation, and examples of the gamma distribution. Gamma distribution is a popular continuous probability distribution that has a variety of applications in various fields such as finance, engineering, physics, and social sciences. It is a flexible distribution that can take on a variety of shapes depending on its parameters. It can often provide a good fit to real-world data and has a simple and well-known probability density function that can be easily computed.