WEIBULL.DIST

Calculates the probability density function (PDF) or the cumulative distribution function (CDF) of the Weibull distribution.

Syntax:

WEIBULL.DIST(x, alpha, beta, cumulative)

x is required, and is the value you want to use to evaluate the distribution.

alpha is required, and is the scale parameter of the distribution.

beta is required, and is the shape parameter of the distribution.

cumulative is required, and is a logical value:

TRUE: Returns the cumulative distribution function (CDF), which gives the probability that a random variable is less than or equal to x.

FALSE: Returns the probability density function (PDF), which gives the probability density at a specific value x.

Example:

If x contains 127, alpha contains 25, beta contains 153 and cumulative contains TRUE:

WEIBULL.DIST(127, 25, 153, TRUE)

returns 0.00945683

x:

alpha:

beta:

Cumulative:

Result:

0.00945683

Application:

Light Bulb Lifespan Analysis

Scenario: A light bulb manufacturer wants to determine the reliability of a new batch of LED light bulbs. They have collected data on the lifespan of these bulbs and want to calculate the probability of a bulb failing at or before a certain number of hours.

The Weibull distribution is a good model for this scenario because it can represent different failure rate patterns:

Decreasing failure rate: If the failure rate decreases over time, it suggests that weak bulbs fail early, and the remaining ones are more robust.
Constant failure rate: This indicates random failures, often associated with external factors.
Increasing failure rate: This is the most common scenario for manufactured products, as parts wear out over time.

Parameters for the WEIBULL.DIST function:

The WEIBULL.DIST(x, alpha, beta, cumulative) function requires the following inputs:

x: The value at which to evaluate the function (e.g., the number of hours).
alpha (shape parameter): This parameter, often denoted by k, describes the shape of the distribution.
- If k<1, the failure rate is decreasing.
- If k=1, the failure rate is constant (exponential distribution).
- If k>1, the failure rate is increasing.
beta (scale parameter): This parameter, often denoted by λ, relates to the characteristic life of the product, where about 63.2% of the products are expected to fail.
cumulative: A logical value (TRUE or FALSE).
- TRUE: Calculates the cumulative distribution function (CDF), which gives the probability that a bulb will fail at or before time x.
- FALSE: Calculates the probability density function (PDF), which gives the relative likelihood of failure at time x.

Data:

Based on previous reliability tests, the manufacturer has determined the following parameters for their LED bulbs:

Shape parameter (α): 2.5 (This suggests an increasing failure rate, as expected for components that wear out)
Scale parameter (β): 20,000 hours (This indicates that about 63.2% of the bulbs are expected to fail by 20,000 hours).

Analysis using WEIBULL.DIST:

The manufacturer wants to calculate the cumulative probability of failure at different time intervals to understand the product's reliability.

Table of Results:

The table below shows the calculated cumulative probability of failure for a range of hours using the formula

WEIBULL.DIST(A1, 2.5, 20000, TRUE).

	Hours (x)	Cumulative Probability of Failure	Interpretation
	A	B	C
1	5,000	0.030766766	There is a 3.07% probability that a bulb will fail at or before 5,000 hours.
2	10,000	0.162033114	By 10,000 hours, the cumulative probability of failure rises to 16.20%.
3	15,000	0.38561855	By 15,000 hours, the probability of failure is nearly 40%.
4	20,000	0.632120559	This is the characteristic life. At 20,000 hours, about 63.2% of the bulbs are expected to have failed. This value is always 1−e−1 regardless of the shape parameter.
5	25,000	0.825691421	By 25,000 hours, a large majority of the bulbs are expected to have failed.
6	30,000	0.936433963	The vast majority of the bulbs are expected to fail by 30,000 hours.

Conclusion

This analysis, using the WEIBULL.DIST function, allows the manufacturer to make more informed decisions about product warranties, marketing claims, and expected product lifetime. For example, they might decide to offer a warranty of 10,000 hours, knowing that the expected failure rate by that time is 16.20%. This is a much higher rate than the initial estimate and might prompt the company to reconsider its warranty period. The analysis confirms the wear-and-tear nature of the product, with a failure probability that increases significantly over time, as described by the shape parameter, α = 2.5.