# Gamma Distribution Calculator

Use this calculator to compute probabilities, percentiles, and visualize the Gamma distribution. Input the shape and scale parameters, and the specific value(s) to calculate left, right, between, or outside probabilities.

**Calculated Probability: **

## Understanding the Gamma Distribution

The **Gamma distribution** is a continuous probability distribution used to model the time until an event occurs a certain number of times. It is commonly applied in fields like survival analysis, queuing models, and insurance risk.

### Key Components of the Gamma Distribution

**Shape Parameter (k)**: Defines the shape of the distribution. Higher values of \( k \) result in a more symmetric distribution.**Scale Parameter (θ)**: This stretches or compresses the distribution horizontally. It is also important to note that the **rate parameter** \( \gamma \) is the inverse of the scale parameter, meaning \( \gamma = 1/\theta \).

### Gamma Distribution Formula

The probability density function (PDF) for the Gamma distribution is given by the formula:

### Conditions for Using the Gamma Distribution

The Gamma distribution applies under specific conditions:

**Non-Negativity**: The Gamma distribution is only defined for non-negative values, meaning that \( X \geq 0 \).**Shape and Scale Parameters**: The shape parameter \( k \) and the scale parameter \( \theta \) must be positive real numbers.

### Step-by-Step Example: Finding the Probability

Suppose we want to calculate the probability that a random variable \( X \), following a Gamma distribution with shape parameter \( k = 2 \) and scale parameter \( θ = 3 \), takes a value less than \( x = 5 \).

#### Step 1: Formula Setup

The probability we are looking for is \( P(X < 5) \). This is given by the cumulative distribution function (CDF) of the Gamma distribution:

#### Step 2: Plug in the Values

Substituting \( k = 2 \) and \( \theta = 3 \) into the formula, we get:

Here, \( 3^2 = 9 \) and the Gamma function \( \Gamma(2) \) is evaluated as \( \Gamma(2) = (2 - 1)! = 1! = 1 \). The Gamma function \( \Gamma(n) \) is a generalization of the factorial function, where for positive integers, it is equivalent to \( (n-1)! \).

#### Step 3: Compute the Integral

After evaluating the integral, we find:

### Further Reading

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.