# How to use the GAMMA.DIST function

**How to use the GAMMA.DIST function?**

The GAMMA.DIST function calculates the gamma often used in queuing analysis (probability statistics) that may have a skewed distribution. This function was introduced in Excel 2010 and has replaced the GAMMADIST function.

### Table of Contents

## 1. Introduction

**When to use the exponential distribution and when to use the gamma distribution in a poisson process?**

Use the exponential distribution when you want to model the inter-arrival times or waiting times between individual events in a Poisson process. The rate of occurrence of events is constant over time. You are interested in the probability of waiting a certain amount of time until the next event occurs.

Use the gamma distribution when you want to model the total waiting time or the sum of inter-arrival times until a specific number of events occur. The number of events you are interested in is a fixed value (e.g., the total time until the 5th event occurs). You want to analyze the distribution of the total time elapsed until a certain number of events happen.

**What is a gamma distribution?**

The gamma distribution is a two-parameter continuous probability distribution defined for positive values. It generalizes other distributions like chi-square and exponential and models positively skewed data like time durations.

**What is queuing analysis in probability statistics?**

Queuing analysis involves using probability and statistics to analyze and model queues (waiting lines). Studies arrival patterns, service times, queue capacity to describe performance and uses probability distributions like Poisson and exponential distributions. It calculates key metrics like expected wait times, queue lengths, utilization.

**What is a skewed distribution?**

Skewness and asymmetric tails describe the lack of balance and symmetry in probability distributions and datasets. Identifying skewness is fundamental in statistics.

Positive skewness implies a group of values with an asymmetric tail moving toward more positive values. Negative skewness implies a group of values with an asymmetric tail moving toward more negative values.

**What is an asymmetric tail?**

A symmetric distribution like the normal distribution has no skewness. An asymmetric tail is when one tail of the distribution extends further than the other.

## 2. GAMMA.DIST Function Syntax

GAMMA.DIST(*x,alpha,beta,cumulative*)

x |
Required. |

alpha |
Required. |

beta |
Required. 1 returns the standard gamma distribution. |

cumulative |
Required. A boolean value. TRUE - cumulative distribution function FALSE - probability density function |

**What is a density function?**

A density function in statistics describes the relative likelihood that a random variable takes on a given value. The area under the entire density function integrates to 1.

There are two main types of density functions:

- Probability Density Function (PDF)

PDFs are used for continuous random variables and provides the probability that the variable's value lies within a small range around x.

The area under PDF over an interval provides the probability that the variable lies within that interval. - Probability Mass Function (PMF)

PMFs are used for discrete random variables and gives the exact probability that a variable takes on a specific value x.

## 3. Example 1

**In a manufacturing process, the time taken to complete a task follows a gamma distribution. The tasks complete at an average rate of 6 per hour. What is the probability that 5 tasks are completed in less than 30 minutes?**

The arguments in the GAMMA.DIST function are:

x = 30 (minutes)

alpha = 5 (tasks)

beta = 60 minutes / 6 tasks = 10 minutes/task

cumulative = true

Formula in cell C20:

The formula returns a probability value of 0.1847 or 18.47% that 5 tasks are completed in less than 30 minutes.

In the image above, locate the value 30 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the cumulative distribution function. Then, follow the point of intersection horizontally towards the y-axis. You will find that the corresponding value on the y-axis is approximately 0.18.

## 4. Example 2

**In a call center, the time between customer arrivals follows a gamma distribution. The average number of customer arrivals is 12 per hour. What is the probability that the next customer will arrive between 3 and 5 minutes?**

The arguments in the GAMMA.DIST function are:

x = 3 and 5 (minutes)

alpha = 1 (customer)

beta = 60 minutes / 12 customer = 5 minutes/customer

cumulative = true

Formula in cell C20:

The formula returns a probability value of 0.6321 for GAMMA.DIST(5,C17,C18,C19) and 0.4512 for GAMMA.DIST(3,C17,C18,C19). The difference is 0.1809 or 18.1% which is the probability that 1 customer will arrive between 3 and 5 minutes.

In the image above, locate the value 5 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the cumulative distribution function. Then, follow the point of intersection horizontally towards the y-axis. You will find that the corresponding value on the y-axis is approximately 0.6321

Also, locate the value 3 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve. Then, follow the point of intersection horizontally towards the y-axis. You will find that the corresponding value on the y-axis is approximately 0.4512

## 5. GAMMA.DIST Function not working

The GAMMA.DIST function returns

- #VALUE! error value if
*alpha, beta*or*x*is non-numeric. - #NUM! error value if:
- x < 0 (zero)
- alpha <= 0 (zero)
- beta <= 0 (zero)

## 6. How is the GAMMA.DIST Function calculated?

The equation to calculate the gamma probability density function is as follows:

The gamma distribution is defined by two parameters: the shape parameter (α) and the scale parameter (β). The mathematical formula for the gamma distribution is given by:

Probability Density Function (PDF):

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

where:

x is the variable (the value at which you want to calculate the distribution)

α is the shape parameter (α > 0)

β is the scale parameter (β > 0)

Γ(α) is the gamma function, which is an extension of the factorial function to non-integer values

The standard gamma probability function:

### Functions in 'Statistical' category

The GAMMA.DIST function function is one of 73 functions in the 'Statistical' category.

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