BETAINV

Calculates the inverse of the BETADIST function.

Syntax:

BETAINV(p, α, β, a, b)

The beta distribution is a family of continuous probability distributions, defined for an interval a to b, where α and β are parameters controlling the shape of the distribution.

BETAINV returns the value n, lying between a and b, such that BETADIST(n, α, β, a, b) is p.

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

Example:

BETAINV(0.96, 3, 4)

returns approximately 0.75.

Application:

The BETAINV function in statistics is used to calculate the inverse of the cumulative beta probability density function. In simpler terms, if you have a probability value (p) and the parameters of a beta distribution (α and β), BETAINV can tell you the corresponding value (x) from the distribution for which the cumulative probability is equal to p.

This function is particularly useful in project management, specifically in the Program Evaluation and Review Technique (PERT) method. PERT is a statistical tool used to analyze and represent the tasks involved in a project. It uses a beta distribution to model the time it takes to complete a task, considering optimistic, most likely, and pessimistic estimates.

Let's consider a simple example:

A project manager is planning the development of a new software feature. They have a task, "Code Development," and they've gathered three time estimates from the development team:

Optimistic time (a): 15 days (the shortest time the task could take)
Most likely time (m): 25 days (the most probable time)
Pessimistic time (b): 60 days (the longest time the task could take)

In PERT, the parameters for the beta distribution are calculated as follows:

α=1+4( $\frac{m - a}{b - a}$ )
β=1+4( $\frac{b - m}{b - a}$ )

Using these formulas:

α=1+4( $\frac{25 - 15}{60 - 15}$ )=1+4( $\frac{10}{45}$ )=1.889
β=1+4( $\frac{60 - 25}{60 - 15}$ )=1+4( $\frac{35}{45}$ )=4.111

The project manager wants to find the number of days by which there is a 90% probability that the "Code Development" task will be completed. In statistical terms, they want to find the value of x for which the cumulative probability is 0.90. This is a perfect use case for the BETAINV function.

The function call would look like this:

BETAINV(0.90, 1.889, 4.111, 15, 60)

Probability (p): 0.90
Alpha (α): 1.889
Beta (β): 4.111
Lower bound (a): 15
Upper bound (b): 60

Result: The BETAINV function would return a value of approximately 40 days.

This means there is a 90% probability that the "Code Development" task will be completed within 40 days.

Here is a table summarizing the values:

	Parameter	Value	Description
	A	B	C
1	a	15	Optimistic time estimate (lower bound)
2	m	25	Most likely time estimate
3	b	60	Pessimistic time estimate (upper bound)
4	$α$	1.889	Calculated alpha parameter for beta distribution
5	$β$	4.111	Calculated beta parameter for beta distribution
6	p	0.9	Desired cumulative probability (90%)
7	Result	40	Number of days to achieve 90% completion probability

This example shows how the BETAINV function can provide a critical insight for project managers, allowing them to set realistic deadlines and manage stakeholder expectations by understanding the probability associated with project timelines.