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BETADIST

Calculates the cumulative distribution function or the probability density function of a beta distribution.

Syntax

BETADIST(x, α, β, a, b, cumulative)

The beta distribution is a family of continuous probability distributions, where α and β are parameters controlling the shape of the distribution.

x is the number, at which you will evaluate the Beta distribution.

The parameters a and b are lower and upper bounds of the distribution. You can interpret the value a as location and the value b−a as scale.

a and b are optional parameters which default (if omitted) to 0 and 1.

cumulative is an optional, logical parameter which defaults to TRUE() if omitted.

Constraints:

α > 0 , β > 0 , a < b
If α < 1, than the density function has a pole at x = a.
If β < 1, than the density function has a pole at x = b.

Semantic

For cumulative = FALSE() the function BETADIST calculates the probability density function (besides the constraints given above):

$f (x) = {0, \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} \cdot (\frac{x - a}{b - a})^{α - 1} \cdot (1 - \frac{x - a}{b - a})^{β - 1} \cdot \frac{1}{b - a}, if x < a \lor x > b if a \leq x \leq b$

For cumulative = TRUE() the function BETADIST calculates the cumulative distribution function:

$F (x) = ⎩ ⎨ ⎧ 0, 1, \int_{a}^{x} \frac{Γ ( α + β )}{Γ ( α ) Γ ( β )} \cdot (\frac{x - a}{b - a})^{α - 1} \cdot (1 - \frac{x - a}{b - a})^{β - 1} d t, if x < a if x > b if a \leq x \leq b$

Notice, that

$F (x) = I_{z} (α, β)$

where

$z = \frac{x - a}{b - a}$

and $I_{z}$ is the regularized incomplete Beta function.

Example

BETADIST(1,5,3,-2,4,FALSE())returns approximately 0.273

BETADIST(0.2,0.7,4,0,1,FALSE())returns approximately 1.644

BETADIST(1,1,0.5,0,1,FALSE())

returns Invalid argument because there is a pole

BETADIST(1.1,1,0.5,0,1,FALSE())

returns 0

BETADIST(1,5,3,-2,4,TRUE())

returns approximately 0.227

BETADIST(0.2,0.7,4,0,1,TRUE())returns approximately 0.718

BETADIST(1.1,1,0.5,0,1,TRUE())returns 1

Application:

Project Completion Time

Scenario: A project manager wants to estimate the probability of completing a new software feature within a certain timeframe. Based on past projects, the completion time can be modeled using a beta distribution. The project manager has determined the following parameters:

Alpha (α): 2.5 (Represents the shape of the distribution, influenced by factors like favorable conditions or skilled team members)
Beta (β): 4.0 (Represents the shape of the distribution, influenced by factors like potential delays or unexpected challenges)
Minimum Time (A): 0 days (The earliest possible completion time)
Maximum Time (B): 30 days (The latest possible completion time, representing a hard deadline)

The project manager wants to find the probability of completing the feature within 15 days.

Using the BETADIST function:

The BETADIST function can be used to calculate this.

Formula: BETADIST(x, alpha, beta, A, B)

Where:

x: The value for which you want to find the cumulative probability (15 days)
alpha: The alpha parameter (2.5)
beta: The beta parameter (4.0)
A: The minimum value (0)
B: The maximum value (30)

Calculation:

BETADIST(15, 2.5, 4.0, 0, 30)

Result: Approximately 0.736

Interpretation: This means there is an 73.6% probability that the project will be completed within 15 days.

Table:

	Argument	Value	Description
	A	B	C
1	x	15	The specific number of days for which we want to calculate the cumulative probability.
2	Alpha (α)	2.5	A parameter representing the shape of the distribution.
3	Beta (β)	4	A parameter representing the shape of the distribution.
4	A	0	The minimum possible number of days for project completion.
5	B	30	The maximum possible number of days for project completion.

This example shows how the BETADIST function can be a valuable tool for risk analysis and decision-making in project management by providing a quantifiable probability of success within a defined timeframe.

Result for BETADIST(15, 2.5, 4.0, 0, 30):

73.6%