Use this gamma distribution calculator to quickly and accurately compute the probability and cumulative values for different parameters.

## How to Use the Gamma Distribution Calculator

To use this calculator, please follow the steps below:

- Enter the Shape (α) parameter. This should be a positive number.
- Enter the Scale (β) parameter. This should be a positive number.
- Enter the X value at which you want to evaluate the PDF (Probability Density Function) and CDF (Cumulative Distribution Function).
- Optionally, enter the lower and upper bounds for the CDF calculation. The default lower bound is 0 and the upper bound is 5.
- Click on the “Calculate” button.

## How It Calculates the Results

The calculator computes the gamma distribution based on the formula for the Gamma PDF:

f(x; α, β) = (x^(α-1) * e^(-x/β)) / (Γ(α) * β^α)

where `Γ(α)` is the gamma function of the shape parameter α.

The CDF is computed by integrating the PDF over the range from the lower bound to the upper bound.

## Limitations

– The calculator assumes that all parameters provided are valid. Ensure that you enter positive values for α and β.

– The integration for the CDF is done using a simple numerical method, which may not be extremely accurate for very small or very large values.

– This calculator does not handle negative x values for the gamma distribution as it is not defined for negative x values.

– The calculations may become less accurate for very large values of the shape parameter α.

## Use Cases for This Calculator

### Calculate Probability Density Function (PDF)

You can use this feature to calculate the Probability Density Function (PDF) of a gamma distribution by entering the shape and rate parameters. The PDF shows how likely a random variable falls within a particular range, providing insights into the distribution’s behavior.

### Calculate Cumulative Distribution Function (CDF)

With this option, you can determine the Cumulative Distribution Function (CDF) of a gamma distribution by inputting the shape and rate parameters. The CDF represents the probability that a random variable is less than or equal to a certain value, aiding in understanding the distribution’s cumulative behavior.

### Generate Probability Values

You have the ability to generate probability values for specific points in a gamma distribution using this function. By entering the shape and rate parameters along with a value, you can quickly get the probability of the random variable being less than or equal to that value.

### Calculate Percentiles

Use this tool to calculate percentiles for a gamma distribution by providing the shape and rate parameters along with a percentile value. Percentiles give you insight into relative standing and allow you to compare values within the distribution.

### Estimate Moments

Estimate moments such as mean, variance, skewness, and kurtosis for a gamma distribution using this feature. Moments provide information about the shape and behavior of the distribution, aiding in statistical analysis and model building.

### Compute Quantiles

You can compute quantiles for a gamma distribution by entering the shape and rate parameters along with a probability value. Quantiles help in understanding the spread and variability of the distribution, making it easier to interpret data.

### Calculate Survival Function

Determine the Survival Function for a gamma distribution by providing the shape and rate parameters. The Survival Function gives the probability that a random variable is greater than a certain value, providing valuable insights for reliability and survival analysis.

### Plot Probability Density Function

Visualize the Probability Density Function (PDF) of a gamma distribution by plotting it based on the shape and rate parameters. This graphical representation helps in understanding the shape and characteristics of the distribution.

### Plot Cumulative Distribution Function

Generate a plot for the Cumulative Distribution Function (CDF) of a gamma distribution by entering the shape and rate parameters. The plot shows how the cumulative probability changes with different values, aiding in interpretation and decision-making.

### Compare Multiple Distributions

Compare multiple gamma distributions by entering their respective shape and rate parameters to analyze and contrast their PDFs, CDFs, percentiles, and other statistical metrics. This feature is useful for academic research, data analysis, and decision support.