The Weibull distribution is a continuous probability distribution that is commonly used in reliability engineering and statistical analysis.

In addition to the two-parameter Weibull distribution, there is also a three-parameter Weibull distribution. The third parameter is the location parameter, also known as the threshold parameter, which determines the point at which the distribution begins. When the threshold parameter is zero, the three-parameter Weibull distribution reduces to the two-parameter Weibull distribution.

## Probability Density Function (PDF)

The probability density function (PDF) of the three-parameter Weibull distribution is given by:

$$f(x) = \frac{\alpha}{\beta} \left(\frac{x-\gamma}{\beta}\right)^{\alpha-1} e^{-\left(\frac{x-\gamma}{\beta}\right)^\alpha}$$

## Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the three-parameter Weibull distribution is given by:

$$F(x) = 1 - e^{-\left(\frac{x-\gamma}{\beta}\right)^\alpha}$$

Where alpha (α) is the shape parameter, beta (β) is the scale parameter, and gamma (γ) is the location parameter.

The three-parameter Weibull distribution is often used to model failure times that have a threshold, such as the time it takes for a device to fail after it has been turned on.

## Mean and Variance:

The mean and variance of the three-parameter Weibull distribution can be calculated using the formulas:

**Mean:**

$$\mu = \beta\Gamma\left(1 + \frac{1}{\alpha}\right) + \gamma$$

**Variance:**

$$\sigma^2 = \beta^2\left[\Gamma\left(1 + \frac{2}{\alpha}\right) - \Gamma^2\left(1 + \frac{1}{\alpha}\right)\right]$$

Where beta is the scale parameter, alpha is the shape parameter, gamma is the location parameter, and Gamma is the gamma function.

## Using Excel

WEIBULL.DIST(x–γ, β, α, cum)

There is no separate formula for 3-parameter Weibull Distribution in Excel. However, you can use the 2-parameter Weibull Distribution formula with the location parameter (γ) as an additional argument.

The syntax for this is WEIBULL.DIST(x–γ, β, α, cum).

Where x is the value for which you want to calculate the probability, γ is the location parameter, β is the scale parameter, and α is the shape parameter. The last argument, “cum ” is a logical value that determines whether the function returns the cumulative distribution function (CDF) or the probability density function (PDF).

If cum is TRUE, WEIBULL.DIST returns the CDF; if FALSE, it returns the PDF.

## Summary:

In summary, the three-parameter Weibull distribution is useful for modelling failure times and other phenomena that exhibit a threshold. It has three parameters (alpha, beta, and gamma) that can be adjusted to fit the data and make predictions.