# Gamma Distribution Calculator

## StatsCalculator.com

### Related Tools

Gamma PDF: 0.0895

P(X <= k): 0.3554

P(X >= k): 0.6446

## Gamma Distribution Calculator: Understanding the Probability Density Function and Related Calculators for Skewed Distributions

A gamma distribution calculator is a tool used to calculate probabilities for the gamma distribution, a continuous probability distribution widely used in statistics. The gamma distribution is defined by two parameters: the shape parameter k and the scale parameter b. The probability density function and cumulative distribution function of the gamma distribution can be calculated using the gamma function, which is a generalization of the factorial function.

The gamma distribution is used to model a wide range of phenomena, including waiting times, failure rates, and income distribution. It is related to other probability distributions such as the exponential distribution, Poisson distribution, and beta distribution. The gamma distribution is a skewed distribution, with its shape determined by the value of the shape parameter k. The scale parameter b determines the spread of the distribution and its variance. Confidence intervals and quantiles can be calculated using the distribution function and quantile function, respectively.

A gamma distribution calculator can be used to solve a variety of problems in mathematics and statistics. For example, it can be used to calculate the probability of a given value for a random variable following a gamma distribution, or to find the appropriate value of the shape parameter k or scale parameter b for a given sample size and desired level of confidence. Other related calculators include those for the standard gamma distribution, beta distribution, Erlang distribution, and incomplete gamma function.

## Gamma Distribution Calculator

### What is a Gamma Distribution?

A gamma distribution is a continuous probability distribution that is commonly used in statistics to model positive, skewed data. It is a two-parameter family of distributions that includes the exponential and chi-squared distributions as special cases. The gamma distribution is defined by two parameters: the shape parameter k and the scale parameter b.

A gamma distribution calculator is a tool that allows users to calculate various properties of a gamma distribution, including the probability density function, cumulative distribution function, quantile function, and more. These calculators are useful for researchers, statisticians, and anyone working with data that follows a gamma distribution.

### Gamma Function

The gamma function is a mathematical function that is used to define the gamma distribution. It is defined as the integral of x^(k-1)*e^(-x)dx from 0 to infinity, where k is the shape parameter of the gamma distribution. The gamma function is denoted by the symbol Γ(k).

### Probability Density Function

The probability density function (PDF) of a gamma distribution is given by the formula f(x) = (x^(k-1)e^(-x/b))/(b^kΓ(k)), where x is the random variable, k is the shape parameter, and b is the scale parameter. The PDF describes the probability of a given value of x occurring in the distribution.

### Cumulative Distribution Function

The cumulative distribution function (CDF) of a gamma distribution is the probability that a random variable X is less than or equal to a given value x. The CDF is given by the formula F(x) = Γ(k,x/b)/Γ(k), where Γ(k,x/b) is the incomplete gamma function.

### Quantile Function

The quantile function of a gamma distribution is the inverse of the CDF. It gives the value of x for a given probability α. The quantile function is denoted by the symbol Q(α) and is given by the formula Q(α) = b*Γ(k,α)/Γ(k).

### Shape and Scale Parameters

The shape parameter k and the scale parameter b are the two parameters that define the gamma distribution. The shape parameter determines the shape of the distribution, while the scale parameter determines the scale. The mean and variance of a gamma distribution are given by E(X) = kb and Var(X) = k*b^2.

### Standard Gamma Distribution

The standard gamma distribution is a special case of the gamma distribution where the scale parameter b is equal to 1. The PDF of the standard gamma distribution is given by f(x) = x^(k-1)*e^(-x)/Γ(k).

### Related Calculators

There are several related calculators that are useful for working with the gamma distribution, including calculators for the Poisson distribution, beta distribution, Erlang distribution, and more. These calculators can help researchers and statisticians to better understand the properties of the gamma distribution and its relationship to other probability distributions.

In summary, the gamma distribution calculator is a useful tool for anyone working with data that follows a gamma distribution. By understanding the properties of the gamma distribution, including the gamma function, probability density function, cumulative distribution function, and quantile function, researchers and statisticians can better analyze and interpret their data.

## Properties of Gamma Distribution

Gamma distribution is a continuous probability distribution that is widely used in statistics. It is a two-parameter family of probability distribution that includes the exponential and chi-squared distributions as special cases. In this section, we will explore some of the properties of gamma distribution.

### Mean and Variance

The mean and variance of gamma distribution depend on its shape and scale parameters. The mean of gamma distribution is given by the formula:

`mean = k * b`

where `k` is the shape parameter and `b` is the scale parameter. The variance of gamma distribution is given by the formula:

`variance = k * b^2`

### Mode

The mode of gamma distribution is given by the formula:

`mode = (k - 1) * b`

for `k > 1`. When `k <= 1`, the mode is undefined.

### Skewed Distribution

Gamma distribution is a skewed distribution, which means that it is not symmetric around its mean. The skewness of gamma distribution depends on its shape parameter `k`. When `k < 1`, the distribution is highly skewed to the right. When `k > 1`, the distribution is skewed to the left. When `k = 1`, the distribution is symmetric.

Gamma distribution is related to other probability distributions such as the exponential distribution, Poisson distribution, beta distribution, and Erlang distribution. There are several related calculators that can be used to calculate various properties of gamma distribution such as the density function, cumulative distribution function, quantile function, and incomplete gamma function.

In summary, gamma distribution is a versatile probability distribution that is widely used in statistics and mathematics. Its properties such as mean, variance, mode, and skewness make it a useful tool for modeling various phenomena in science and engineering.

## Parameter Estimation

Parameter estimation is an important aspect of probability distribution. It involves the estimation of unknown parameters of a probability distribution from a given set of data. In the case of the gamma distribution calculator, the parameters to be estimated are the shape parameter k and the scale parameter b.

### Method of Moments

The method of moments is a parameter estimation technique that involves equating the sample moments to the corresponding population moments. For the gamma distribution, the method of moments estimator for the shape parameter k is given by:

k = (mean^2) / variance

where mean and variance are the sample mean and variance, respectively. The method of moments estimator for the scale parameter b is given by:

b = variance / mean

where mean and variance are the sample mean and variance, respectively.

### Maximum Likelihood Estimation

Maximum likelihood estimation is another parameter estimation technique that involves finding the parameter values that maximize the likelihood function. For the gamma distribution, the likelihood function is given by:

L = (b^k / Gamma(k)) * x^(k-1) * e^(-x*b)

where Gamma is the gamma function, x is the given value, and k and b are the shape and scale parameters, respectively.

The maximum likelihood estimator for the shape parameter k is obtained by solving the equation:

digamma(k) - ln(x/b) = ln(mean)

where digamma is the digamma function and mean is the sample mean.

The maximum likelihood estimator for the scale parameter b is obtained by solving the equation:

k/b = mean

where mean is the sample mean.

### Example

Suppose a sample of size n=100 is drawn from a gamma distribution with unknown parameters k and b. The sample mean and variance are found to be 5 and 10, respectively. Using the method of moments, the estimators for k and b are:

k = (5^2) / 10 = 2.5 b = 10 / 5 = 2

Using maximum likelihood estimation, the estimators for k and b are obtained by solving the equations:

digamma(k) - ln(x/b) = ln(mean) => k = 2.67 k/b = mean => b = 1.87

In conclusion, parameter estimation is an essential part of probability distribution, and the gamma distribution calculator provides various methods for estimating the shape and scale parameters of the gamma distribution. The method of moments and maximum likelihood estimation are two widely used techniques for parameter estimation, and both have their advantages and disadvantages.

## Applications of Gamma Distribution

Gamma distribution is a continuous probability distribution that is used to model the probability of a random variable taking on a certain value. It has a probability density function (PDF) that is defined by two parameters: the shape parameter, k, and the scale parameter, b. The gamma distribution calculator is a useful tool that can be used to calculate the gamma distribution, the gamma function, the cumulative distribution function, and the gammadist function.

### Exponential Distribution

The exponential distribution is a special case of the gamma distribution, where the shape parameter, k, is equal to 1. It is commonly used to model the time between events that occur at a constant rate, such as the time between radioactive decay events or the time between customer arrivals at a store. The exponential distribution has a probability density function that is defined by the scale parameter, b, and is given by:

f(x) = (1/b) * e^(-x/b)

### Poisson Distribution

The Poisson distribution is another special case of the gamma distribution, where the shape parameter, k, is equal to 0. It is commonly used to model the number of events that occur in a fixed interval of time or space, such as the number of phone calls received by a call center in an hour or the number of cars that pass through a toll booth in a day. The Poisson distribution has a probability mass function that is defined by the mean, λ, and is given by:

P(X = k) = (e^(-λ) * λ^k) / k!

### Chi-Square Distribution

The chi-square distribution is a special case of the gamma distribution, where the shape parameter, k, is equal to n/2 and the scale parameter, b, is equal to 2. It is commonly used to test the goodness of fit of a sample to a theoretical distribution or to test the independence of two categorical variables. The chi-square distribution has a probability density function that is defined by the degrees of freedom, n, and is given by:

f(x) = (1/(2^(n/2) * Γ(n/2))) * x^(n/2-1) * e^(-x/2)

### Beta Distribution

The beta distribution is a continuous probability distribution that is used to model the probability of a random variable taking on a value between 0 and 1. It has two shape parameters, α and β, that control the shape of the distribution. The beta distribution is commonly used in Bayesian statistics to model the posterior distribution of a parameter. The beta distribution has a probability density function that is defined by the shape parameters, α and β, and is given by:

f(x) = (x^(α-1) * (1-x)^(β-1)) / (B(α,β))

where B(α,β) is the beta function.

### Erlang Distribution

The Erlang distribution is a special case of the gamma distribution, where the shape parameter, k, is an integer. It is commonly used to model the time between events that occur at a constant rate, but with a minimum number of events required before the time interval is considered. The Erlang distribution has a probability density function that is defined by the shape parameter, k, and the scale parameter, b, and is given by:

f(x) = (x^(k-1) * e^(-x/b)) / (b^k * (k-1)!)

### Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is commonly used to model the distribution of a random variable. It has two parameters, the mean, μ, and the variance, σ^2, that control the shape of the distribution. The normal distribution has a probability density function that is defined by the mean, μ, and the variance, σ^2, and is given by:

f(x) = (1/(σ * sqrt(2π))) * e^(-(x-μ)^2/(2σ^2))

In conclusion, the gamma distribution calculator is a useful tool for calculating the gamma distribution, the gamma function, and other related calculators. The gamma distribution has many applications in statistics, including the exponential distribution, Poisson distribution, chi-square distribution, beta distribution, Erlang distribution, and normal distribution. By understanding these applications, one can better understand the role of the gamma distribution in mathematics and statistics.

## Examples and Solutions

### Example Problem

Suppose a company wants to estimate the time it takes to complete a certain task. The company has data from previous tasks and has determined that the time follows a gamma distribution with shape parameter k = 3 and scale parameter b = 2.5. The company wants to know the probability that the task will be completed in less than 8 hours.

### Step-by-Step Procedure

To solve this problem using the gamma distribution calculator, follow these steps:

1. Go to the gamma distribution calculator website.
2. Enter the shape parameter k as 3 and the scale parameter b as 2.5.
3. Select the cumulative distribution function option.
4. Enter the given value of 8 hours as the value of x.
5. Click on the calculate button to get the probability.

### Calculator Solution

Using the gamma distribution calculator with k = 3 and b = 2.5, the probability that the task will be completed in less than 8 hours is 0.967.

The probability density function for the gamma distribution is given by the formula:

f(x) = (x^(k-1) * e^(-x/b)) / (b^k * Gamma(k))

where Gamma(k) is the gamma function.

The gamma distribution is a continuous probability distribution that is used to model the waiting time until a specified number of events occur. It is a generalization of the exponential distribution and is related to the Poisson distribution.

The gamma distribution has two parameters: the shape parameter k and the scale parameter b. The shape parameter determines the shape of the density function, while the scale parameter determines the spread of the distribution.

The gammadist function in Excel can be used to calculate the gamma distribution for a given value of x, shape parameter k, and scale parameter b.

The standard gamma distribution has k = 1 and b = 1. The gamma distribution reduces to the exponential distribution when k = 1. When k is an integer, the gamma distribution is related to the factorial function.

The gamma distribution is a skewed distribution, meaning that it is not symmetric around its mean. The variance of the gamma distribution is k * b^2.

The confidence interval for the gamma distribution can be calculated using the quantile function. The beta parameter is used in the beta distribution, which is a generalization of the gamma distribution. The incomplete gamma function is used to calculate the gamma distribution for non-integer values of k.

The Erlang distribution is a special case of the gamma distribution, where k is an integer and represents the number of events that must occur before the waiting time is measured.

In summary, the gamma distribution calculator is a useful tool for calculating probabilities and other statistics related to the gamma distribution. It is important to understand the parameters and properties of the gamma distribution to use the calculator effectively.

# Conclusion

In conclusion, the gamma distribution calculator is a valuable tool for those working with continuous probability distributions. By inputting values for the shape and scale parameters, users can quickly calculate the probability density function, cumulative distribution function, and quantile function for the gamma distribution.

Additionally, the calculator can be used to calculate related statistics such as the mean, variance, and confidence interval for a given sample size. The gamma distribution is a versatile distribution that can be used to model a wide range of phenomena, including the time to failure of mechanical systems and the size of insurance claims.

While the gamma distribution is often used in conjunction with the exponential distribution and the Poisson distribution, it is important to note that it is a distinct probability distribution with its own unique properties. The gamma function, which is used to define the gamma distribution, is a generalization of the factorial function and plays an important role in many areas of mathematics.

Overall, the gamma distribution calculator is a powerful tool for anyone working with probability distributions and related calculations. Whether you are a mathematician, statistician, or simply someone who needs to perform calculations related to the gamma distribution, this calculator can save you time and help you make more informed decisions based on the data at hand.