# How to use the BETA.DIST function

**What is the BETA.DIST function?**

The BETA.DIST function calculates the beta distribution. This function was introduced in Excel 2010.

#### Table of Contents

## 1. Introduction

**What is the beta distribution?**

The beta distribution is a continuous probability distribution defined over the interval [0, 1] and parameterized by two positive shape parameters, alpha (α) and beta (β).

**What is a continuous probability distribution?**

A continuous probability distribution is a function defined over a range of continuous values that provides the probability of a random variable falling between any two points, having a density described by an equation rather than discrete probabilities.

## 2. Syntax

BETA.DIST(*x,alpha,beta,cumulative,[A],[B]*)

x |
Required. |

alpha |
Required. A parameter which determines the shape of the distribution. |

beta |
Required. A parameter which determines the shape of the distribution. |

cumulative |
Required. A boolean value that determines the form of the function. TRUE - Cumulative beta probability distribution function FALSE - Beta probability density function |

[A] |
Optional. Lower bound, default value 0 (zero). |

[B] |
Optional. Upper bound, default value 1. |

* What is alpha and beta in a beta distribution?*The beta distribution is very flexible due to how alpha and beta shape its variance and skew. Changing their values gives a wide range of distribution shapes. In the beta distribution, alpha and beta are shape parameters that control the form of the distribution:

Alpha (α): Controls the height of the peak. Higher alpha = taller, more concentrated peak. Alpha mainly changes the peak height.

Beta (β): Controls the tails of the distribution. Higher beta = shorter, thinner tails. Beta mainly changes the tail thickness.

As α & β > 1, shape becomes more symmetric and bell-shaped.

As α & β < 1, shape becomes asymmetrical and J or U shaped.

When α = β = 1, the distribution becomes uniform.

They allow modeling different levels of variance and asymmetry.

**What is a cumulative beta probability distribution?**

The cumulative beta distribution function gives the probability that a beta-distributed random variable with parameters α and β will be less than or equal to a given value x, providing the accumulated area under the probability density curve from 0 to x.

**What is a beta probability density distribution?**

A beta probability density distribution is a function whose shape over [0,1] depends on parameters α and β that gives the relative likelihood of a beta-distributed random variable occurring at different points, whose total area under the curve integrates to 1.

**When to use the beta distribution?**

The beta distribution is used to model random variables limited to intervals of 0 to 1, such as binomial success probabilities, percentage or fraction outcomes, and measurements constrained between limits, making it useful in Bayesian statistics, experimental design, weather forecasting, and other applications.

**What are continuous values?**

Continuous values are numbers that can take on any quantity within a range and can have infinitely many possibilities, unlike discrete values which have distinct separated values; continuous values can use intervals and ranges to describe events rather than fixed outcomes.

**What are discrete probabilities?**

Discrete probabilities are individual separated probabilities assigned to each of a countable number of possible outcomes that sum to 1, like rolling a die where each number has its own exact probability, as opposed to continuous distributions.

**What are binomial success probabilities?**

Binomial success probabilities describe the chance of a certain number of “successes” occurring in a fixed number of independent binary trial events modeled by the binomial distribution, like the probability of getting 3 heads in 10 coin flips.

**What is Bayesian statistics?**

Bayesian statistics is an approach to statistics using Bayes' theorem where prior beliefs about probabilities are updated as new evidence is acquired to determine conditional probabilities and update understanding of likelihood.

## 3. Example 1

The BETA.DIST function calculates the cumulative beta distribution representing an outcome in the form of probability between 0 and 1. It can also calculate the probability density function, the third argument lets you select which you want to calculate. The beta distribution is often used to model the uncertainty about the value of a probability or proportion when there is some prior information available.

Alpha is the number of successes plus one and beta is the number of failures plus one. Successes and failures is a generalization, it can be satisfied customers vs not satisfied customers or what ever you want.

**A company manufactures a new product. The first 10 products has 7 working and 3 faulty. Assuming that the proportion of working products follows a beta distribution with parameters α = 8 and β = 4, what is the probability that more than 90% are working for the next products in line?**

Alpha is the number of working products plus one and beta is the number of faulty products plus one. The ratio between the number of working products and the total number of products is 7/10 equals 0.7 The blue line has it's highest point at x-axis value 0.7.

The image above has argument

- x in cell C18 and that value is 0.9
- alpha in cell C19
- beta in cell C20
- the cumulative argument is TRUE

Formula in cell C24:

The formula in cell C24 returns approx. 0.981 which is the cumulative beta distribution meaning the area below the blue density probability function in the chart above up to x value 0.9

The value we are looking for is the area to the right of 0.9 which we can calculate by subtracting 0.947 with 1. 1 - 0.947 equals approx. 0.0185 or 1.85%

The probability that more than 90% are working is 1.85% The orange line in the chart displayed in the image above shows the cumulative value. As time goes by and more and more products are manufactured you can update the parameters of the beta distribution model to get a better probability value.

## 4. Example 2

**This example continues on example 1 above. The company has now manufactured 100 products, 90 working and 10 defect. What is the probability that more than 90% are working for the next products in line?
**

The BETA.DIST function returns approx 0.570 which gives us 1 - 0.570 equals 0.430 or 43 %. This number 43% is the probability that the products to be manufactured in the future have 90% working and 10% defect.

This example shows that as the number of observations increases the probability density curve (blue) gets more narrow meaning the uncertainty is also decreasing. In other words, the function gets better at predicting the probability as the number of observations increases.

## 5. Example 3

**How is the upper [B] and lower [A] arguments related to the percentage value?**

The optional arguments [A] and [B] are lower and upper limits. The BETADIST function uses these limits to assist you calculating the percentage value for you which is a number between 0 and 1.

You will get the same result if you calculate the percentage yourself, here is an example.

The arguments are:

- x = 30
- alpha = 90
- beta = 110
- [A] = 15
- [B] = 45

Formula in cell C7:

The formula in cell C7 returns 0.922 which is 92.2 %. To calculate the percentage (x) we calculate the ratio between x and the total of the upper and lower limit.

30/(15+45) equals 30/60 = 0.5 We can use this number in the BETADIST function without the upper and lower limit and check if we get the same result.

=BETA.DIST(0.5,90,110,TRUE)

returns 0.921957262116145 which is the exact same value as above (0.921957262116145).

## 6. The function is not working?

The BETADIST function returns

- #VALUE! error value if any argument is non-numeric.
- #NUM! error value if:
- alpha <= 0
- beta <= 0
- x < A
- x >B
- A = B

## 7. How is the function calculated mathematically?

The general equation to calculate the beta distribution:

The text representation of the beta distribution math formula is: x^{(α-1)}(1 - x)^{(β-1)} / B(α - β)

Where:

- x is the variable
- α and β are the shape parameters
- B(α - β) is the beta function evaluated at (α - β)

Useful resources

### Functions in 'Statistical' category

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

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