GAMMADIST

Calculates values for a gamma distribution.

Syntax:

GAMMADIST(x, α, β, mode)

The gamma distribution is a family of continuous probability distributions, with parameters α (shape) and β (scale).

If mode is 0, GAMMADIST calculates the probability density function of the gamma distribution:

$\frac{( \frac{x}{β} ) ^{α - 1} e ^{- \frac{x}{β}}}{β Γ ( α )}$

If mode is 1, GAMMADIST calculates the cumulative distribution function of the gamma distribution:

$\int_{0}^{x} \frac{( \frac{t}{β} ) ^{α - 1} e ^{- \frac{t}{β}}}{β Γ ( α )} d t$

x must be greater than or equal to 0.

Example:

GAMMADIST(2, 1, 1, 1)

returns approximately 0.86.

Application:

Let's consider a scenario involving the lifespan of a particular type of light bulb. We have determined that the lifespan of these light bulbs follows a gamma distribution with a shape parameter (α) of 2 and a scale parameter (β) of 500 hours. We can use the GAMMADIST function to calculate the probability of a light bulb failing at different points in time.

The GAMMADIST function has the following syntax:

GAMMADIST(x, alpha, beta, cumulative)

Where:

x: The value at which to evaluate the function (in this case, time in hours).
alpha: The shape parameter.
beta: The scale parameter.
cumulative: A logical value that determines the form of the function. TRUE returns the cumulative distribution function (CDF), which is the probability that the random variable is less than or equal to x. FALSE returns the probability density function (PDF), which is the probability of the random variable being exactly x.

Example:

We want to find the probability of a light bulb failing at exactly 700 hours (PDF) and the probability of a light bulb failing on or before 1000 hours (CDF).

Input values:

alpha = 2
beta = 500

Calculations:

PDF for x = 700: GAMMADIST(700, 2, 500, FALSE) = 0.00069
CDF for x = 1000: GAMMADIST(1000, 2, 500, TRUE) = 0.593994

Table of Calculations:

	x (Time in Hours)	GAMMADIST(x, 2, 500, FALSE)	GAMMADIST(x, 2, 500, TRUE)	Interpretation
	A	B	C	D
1	500	0.000736	0.264241	Probability of failure at exactly 500 hours is low. The cumulative probability of failing on or before 500 hours is about 26.4%.
2	700	0.00069	0.408167	Probability of failure at exactly 700 hours is also low. The cumulative probability of failing on or before 700 hours is about 40.8%.
3	1000	0.000541	0.593994	Probability of failure at exactly 1000 hours is very low. The cumulative probability of failing on or before 1000 hours is about 59.4%.
4	1500	0.000299	0.800852	The cumulative probability of a bulb failing on or before 1500 hours is about 80.1%.

As you can see from the table, the GAMMADIST function allows us to model the time-to-failure for the light bulbs. The PDF values give us the relative likelihood of failure at a specific point in time, while the CDF values tell us the probability of failure occurring within a given time frame. This information is crucial for making informed decisions about maintenance schedules, warranty periods, and product reliability.