The Gamma distribution in R Programming Language is defined as a two-parameter family of continuous probability distributions which is used in exponential distribution, Erlang distribution and chi-squared distribution.
Mathematical Implementation of the Gamma Distribution
A random variable
f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \quad \text{for } x > 0, \alpha, \beta > 0
where:
\alpha > 0 is the shape parameter.\beta > 0 is the rate parameter (sometimes the scale parameter\theta = 1/\beta is used).\Gamma(\alpha) is the gamma function, defined as\Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} dt
Key properties:
- Mean:
\mu = \frac{\alpha}{\beta} - Variance:
\sigma^2 = \frac{\alpha}{\beta^2}
Implementation of Gamma Distribution Functions in R
We will implement various gamma distribution functions available in the R programming language and understand their interpretation through visualizations.
1. dgamma() Function
The dgamma() function in R calculates the probability density function (PDF) of the Gamma distribution at specified values.
Syntax:
dgamma(x_dgamma, shape)
Parameters:
x_dgamma: A numeric vector of values at which to evaluate the gamma density.shape: The shape parameter (α\alphaα) of the Gamma distribution (must be positive).
Example :
We generate a sequence of values from 0 to 2 and compute the gamma probability density function (PDF) with a shape parameter of 6 at those points. Finally, we plot the computed gamma density values to visualize the distribution.
x_dgamma <- seq(0, 2, by = 0.04)
y_dgamma <- dgamma(x_dgamma, shape = 6)
plot(y_dgamma)
Output :
2. pgamma() Function
The pgamma() function computes the cumulative distribution function (CDF) of the Gamma distribution at specified values. It returns the probability that a Gamma distributed random variable is less than or equal to each value in the input.
Syntax:
pgamma(x_pgamma, shape)
Parameters:
- x_pgamma: defines gamma function
- shape: gamma density of input values
Example:
We create a sequence of values from 0 to 2 and calculate the gamma cumulative distribution function (CDF) with shape parameter 6 for those values. Then, we plot the CDF values to visualize the accumulation of probability.
x_pgamma <- seq(0, 2, by = 0.04)
y_pgamma <- pgamma(x_pgamma, shape = 6)
plot(y_pgamma)
Output:
3. qgamma() Function
The qgamma() function computes the quantile function (inverse cumulative distribution function) of the Gamma distribution. It returns the value
Syntax:
qgamma(x_qgamma, shape)
Parameters:
- x_qgamma: defines gamma function
- shape: gamma density of input values
Example:
We generate a sequence of probabilities from 0 to 1 and compute the gamma quantile function (inverse CDF) with a shape parameter of 6 at those probabilities. The resulting quantiles are then plotted to visualize the relationship between cumulative probabilities and their corresponding gamma distribution values.
x_qgamma <- seq(0, 1, by = 0.03)
y_qgamma <- qgamma(x_qgamma, shape = 6)
plot(y_qgamma)
Output:
4. rgamma() Function
The rgamma() function generates random numbers drawn from a Gamma distribution. It is commonly used for simulations and modeling when random samples from a Gamma distribution are required.
Syntax:
rgamma(N, shape)
Parameters:
- N: gamma distributed values
- shape: gamma density of input values
Example:
We set a seed for reproducibility and generate 800 random values from a gamma distribution with shape parameter 5. A histogram with 500 bins is plotted to visualize the distribution of these random gamma values.
set.seed(1200)
N <- 800
y_rgamma <- rgamma(N, shape = 5)
hist(y_rgamma, breaks = 500, main = "")
Output:
In this article, we explored the key functions of the Gamma distribution in R such as dgamma(), pgamma(), qgamma() and rgamma(). We demonstrated how to use these functions for density calculation, cumulative probabilities, quantile estimation and random number generation.
