WEIBULL

Calculates values for a Weibull distribution.

Syntax:

WEIBULL(x, k, λ, mode)

The Weibull distribution is a continuous probability distribution, with parameters k > 0 (shape) and λ > 0 (scale).

If mode is 0, WEIBULL calculates the probability density function of the Weibull distribution:

$\frac{k}{λ} (\frac{x}{λ})^{k - 1} e^{- (x / λ)^{k}}$

If mode is 1, WEIBULL calculates the cumulative distribution function of the Weibull distribution:

$1 - e^{- (x / λ)^{k}}$

Example:

WEIBULL(2.5, 3, 4, 1)

returns approximately 0.2166, the value of the probability density function with k = 3 and λ = 4 at x = 2.5.

WEIBULL(2.5, 3, 4, 0)

returns approximately 0.2295, the value of the cumulative distribution function with k = 3 and λ = 4 at x = 2.5.

Application:

Time-to-Failure for Jet Engine Blades

A jet engine blade manufacturer uses a spreadsheet program to analyze the reliability of their product. The program has a function WEIBULL(x, k, λ, mode) to calculate Weibull distribution values.

The parameters remain the same:

x: Operating Time (in hours), the value at which to evaluate the function.
k: Shape parameter, which is 2.5.
λ: Scale parameter, which is 5000 hours.
mode: A logical value that determines the function's output.
- If mode = 1, it calculates the cumulative distribution function (CDF), representing the probability of failure.
- If mode = 0, it calculates the probability density function (PDF).

The manufacturer wants to create a table to see the probability of failure and the probability of survival at different time intervals.

They will use the WEIBULL function with mode=1 for the cumulative probability of failure.

Summary Table of Probabilities with WEIBULL Function Calls

Weibull Parameters:

k (Shape): 2.5
λ (Scale): 5000 hours

	Operating Time (x) (Hours)	WEIBULL Function Call (for CDF)	Cumulative Probability of Failure, F(x)	Reliability, R(t)=1−F(t)	Interpretation
	A	B	C	D	E
1	1000	WEIBULL(1000, 2.5, 5000, 1)	0.017729494	0.982270506	1% of blades are expected to fail by this time.
2	3000	WEIBULL(3000, 2.5, 5000, 1)	0.24335024	0.75664976	A 24.3% chance of failure by this time.
3	5000	WEIBULL(5000, 2.5, 5000, 1)	0.632120559	0.367879441	The characteristic life, where 63.2% have failed.
4	6000	WEIBULL(6000, 2.5, 5000, 1)	0.793497129	0.206502871	Over 80% of blades are expected to have failed.
5	8000	WEIBULL(8000, 2.5, 5000, 1)	0.960764461	0.039235539	Only a 4% chance of survival; replacement is critical.

This example illustrates how the Weibull function is used in a real-world setting to model product lifespan, predict failure probabilities, and inform critical business decisions like warranty policies, maintenance schedules, and design improvements.