Gamma Distribution Calculator

Calculate the PDF and CDF of the Gamma distribution using our easy-to-use gamma distribution calculator.

x	PDF f(x)	CDF F(x)
Enter valid parameters to see table.

Table of PDF and CDF values for different x around the mean.

What is the Gamma Distribution Calculator?

The gamma distribution calculator is a statistical tool used to determine the probability density function (PDF) and cumulative distribution function (CDF) for a given set of parameters of the gamma distribution. The gamma distribution is a two-parameter family of continuous probability distributions, widely used in various fields like queuing theory, climatology, and financial services to model waiting times, rainfall amounts, or insurance claim sizes.

This gamma distribution calculator takes the shape parameter (α) and the scale parameter (β) (or sometimes the rate parameter λ = 1/β), along with a specific value ‘x’, and calculates f(x) and F(x).

Who should use it?

Statisticians, data scientists, engineers, financial analysts, and researchers often use the gamma distribution and a gamma distribution calculator to model events where the waiting time between Poisson distributed events is relevant, or when dealing with skewed continuous data that is non-negative.

Common Misconceptions

A common misconception is confusing the gamma distribution with the normal distribution. While the normal distribution is symmetric, the gamma distribution is typically skewed to the right and is only defined for positive values. Another is mixing up the scale (β) and rate (λ) parameters; our gamma distribution calculator uses the scale parameter β, but the rate λ is simply 1/β.

Gamma Distribution Formula and Mathematical Explanation

The gamma distribution is defined by two parameters: the shape parameter α (alpha, k in some notations) and the scale parameter β (beta, θ in some notations), both of which must be positive (α > 0, β > 0). Sometimes, the rate parameter λ = 1/β is used instead of the scale.

The Probability Density Function (PDF) of the gamma distribution is given by:

f(x; α, β) = (x^α-1 * e^-x/β) / (β^α * Γ(α))   for x ≥ 0

Where:

• x is the random variable (x ≥ 0)
• α is the shape parameter
• β is the scale parameter
• e is Euler’s number (approx. 2.71828)
• Γ(α) is the gamma function evaluated at α. The gamma function is an extension of the factorial function to complex numbers, defined as Γ(z) = ∫₀^∞ t^z-1e^-t dt.

The Cumulative Distribution Function (CDF), which gives the probability that the random variable is less than or equal to x, is:

F(x; α, β) = γ(α, x/β) / Γ(α) = P(α, x/β)

Where γ(α, x/β) is the lower incomplete gamma function, and P(α, x/β) is the regularized lower incomplete gamma function.

Our gamma distribution calculator computes both f(x) and F(x).

Variables Table

Variable	Meaning	Unit	Typical Range
x	The value at which the PDF/CDF is evaluated	Depends on context (e.g., time, amount)	x ≥ 0
α (alpha)	Shape parameter	Dimensionless	α > 0
β (beta)	Scale parameter	Same as x	β > 0
Γ(α)	Gamma function	Dimensionless	Γ(α) > 0 for α > 0
f(x)	Probability Density Function value	1/(unit of x)	f(x) ≥ 0
F(x)	Cumulative Distribution Function value	Dimensionless (probability)	0 ≤ F(x) ≤ 1

Practical Examples (Real-World Use Cases)

Example 1: Waiting Times

Suppose the time (in minutes) between customer arrivals at a service desk follows a gamma distribution with a shape parameter α = 3 and a scale parameter β = 2 minutes. What is the probability that the waiting time until the 3rd arrival is between 4 and 8 minutes?

We need to find F(8; 3, 2) – F(4; 3, 2). Using the gamma distribution calculator:

• For x=8, α=3, β=2: F(8) ≈ 0.7619
• For x=4, α=3, β=2: F(4) ≈ 0.3233

The probability is 0.7619 – 0.3233 = 0.4386, or about 43.86%.

Example 2: Insurance Claims

The size of insurance claims (in thousands of dollars) for a certain type of policy is modeled by a gamma distribution with α = 2 and β = 5. What is the probability that a claim is less than $10,000?

We need F(10; 2, 5). Using the gamma distribution calculator with x=10, α=2, β=5:

• F(10) ≈ 0.5940

So, there’s a 59.4% chance a claim is less than $10,000.

How to Use This Gamma Distribution Calculator

1. Enter Shape (α): Input the shape parameter α. It must be greater than 0.
2. Enter Scale (β): Input the scale parameter β. It must be greater than 0.
3. Enter Value (x): Input the value x for which you want to calculate the PDF and CDF. It must be 0 or greater.
4. Calculate: Click the “Calculate” button or just change the input values. The gamma distribution calculator will automatically update.
5. Read Results: The calculator displays the PDF f(x), CDF F(x), Mean (αβ), and Variance (αβ²) below the inputs. The primary result highlights the CDF value.
6. View Chart and Table: The chart shows the PDF curve, and the table provides PDF and CDF values for various x values around the mean.
7. Reset: Click “Reset” to return to default values.
8. Copy Results: Click “Copy Results” to copy the main outputs to your clipboard.

This gamma distribution calculator provides immediate feedback, making it easy to see how changes in α, β, or x affect the distribution.

Key Factors That Affect Gamma Distribution Results

• Shape Parameter (α): This parameter dictates the shape of the distribution. For α ≤ 1, the PDF is L-shaped or strictly decreasing. For α > 1, the PDF has a mode (peak) and is skewed to the right, becoming more symmetric as α increases (approaching normal distribution for large α, given β is constant).
• Scale Parameter (β): This parameter stretches or compresses the distribution along the x-axis. A larger β results in a more spread-out distribution with a larger mean and variance.
• Value (x): This is the point of interest. The PDF f(x) gives the density at x, and the CDF F(x) gives the cumulative probability up to x.
• Mean (αβ): The average value of the distribution increases with both α and β.
• Variance (αβ²): The spread of the distribution increases with α and more significantly with β.
• Gamma Function Γ(α): The value of the gamma function is crucial for scaling the PDF correctly.

Understanding these factors helps in interpreting the results from the gamma distribution calculator and applying them correctly.

Frequently Asked Questions (FAQ)

Q: What is the gamma distribution used for?
A: It’s used to model non-negative, right-skewed continuous data, such as waiting times, rainfall amounts, insurance claim sizes, and server load.

Q: What’s the difference between shape (α) and scale (β) parameters?
A: α controls the shape (peakedness/skewness), while β controls the spread or scale along the x-axis.

Q: Can the shape parameter α be less than 1?
A: Yes, α must be greater than 0. When 0 < α ≤ 1, the PDF starts at infinity (or is very large near x=0) and decreases.

Q: What is the relationship between the gamma distribution and the exponential distribution?
A: The exponential distribution is a special case of the gamma distribution where the shape parameter α = 1. Our exponential distribution calculator can handle this.

Q: What is the relationship between the gamma distribution and the Chi-squared distribution?
A: The Chi-squared distribution with k degrees of freedom is a special case of the gamma distribution with α = k/2 and β = 2.

Q: How does this gamma distribution calculator handle the gamma function?
A: It uses a numerical approximation (like the Lanczos approximation) to calculate the gamma function Γ(α) and the incomplete gamma function for the CDF.

Q: Can I use a rate parameter (λ) instead of scale (β) with this calculator?
A: This calculator directly uses the scale parameter β. If you have the rate λ, simply use β = 1/λ.

Q: Is the gamma distribution always right-skewed?
A: Yes, for α > 0 and β > 0, the gamma distribution is right-skewed, though it becomes more symmetric as α increases. Explore with our gamma distribution calculator to see this.

Related Tools and Internal Resources

• Beta Distribution Calculator: Useful for modeling probabilities and proportions.
• Normal Distribution Calculator: For symmetric, bell-shaped data.
• Poisson Distribution Calculator: Models the number of events in a fixed interval.
• Exponential Distribution Calculator: Models time between Poisson events (gamma with α=1).
• Weibull Distribution Calculator: Often used in reliability engineering and survival analysis.
• Statistics Basics: Learn fundamental statistical concepts.

Using our gamma distribution calculator alongside these resources can provide a comprehensive understanding of probability distributions.