This tool calculates the probability and quantiles of the Weibull distribution for given shape and scale parameters.

## How to Use the Weibull Distribution Calculator

This calculator computes the probability density function (PDF) of the Weibull distribution. To use the calculator:

- Enter the shape parameter (k).
- Enter the scale parameter (λ).
- Enter the value (x) at which you want to evaluate the PDF.
- Click the “Calculate” button.
- The result will be displayed in the “Result” field.

## How It Works

The Weibull distribution is a continuous probability distribution. The PDF of the Weibull distribution for a shape parameter *k* and scale parameter *λ* at a given point *x* is given by the formula:

f(x; k, λ) = (k / λ) * (x / λ)^{(k-1)} * exp(-(x / λ)^{k})

This formula calculates the likelihood of a random variable *x* given the distribution’s shape and scale parameters.

## Limitations

This calculator only works for positive values of the shape parameter, scale parameter, and value (x). The input values must also be finite numbers. Keep in mind that the precision of the result depends on the accuracy of your input values and the limitations of floating-point arithmetic.

## Use Cases for This Calculator

### Calculate Weibull Probability Density Function

Enter the values for the shape and scale parameters to compute the probability density function for a given input in the Weibull distribution. This will help you visualize the distribution and understand the likelihood of different outcomes occurring.

### Compute Weibull Cumulative Distribution Function

Input the shape and scale parameters along with a specific value to get the cumulative distribution function result. This will show you the probability that a random variable is less than or equal to the given input, aiding in risk assessment and decision-making processes.

### Estimate Weibull Percent Point Function

Enter the shape and scale parameters along with a probability value to estimate the point where that probability is achieved in the Weibull distribution. This allows you to determine critical values corresponding to certain probabilities.

### Find Weibull Hazard Function

Input the shape and scale parameters to calculate the hazard function, which describes how the failure rate changes over time in the Weibull distribution. This information is crucial in reliability analysis and survival predictions.

### Determine Weibull Mean and Variance

Simply input the shape and scale parameters to find the mean and variance of the Weibull distribution. Understanding these statistical measures will provide insights into the central tendency and dispersion of the distribution.

### Generate Weibull Quantiles

Specify the shape and scale parameters along with the desired quantile probabilities to generate quantile values in the Weibull distribution. This helps in establishing thresholds and reference points for analysis and decision-making.

### Compute Weibull Skewness and Kurtosis

Enter the shape and scale parameters to calculate the skewness and kurtosis of the Weibull distribution. These measures indicate the asymmetry and peakedness of the distribution, providing additional insights beyond mean and variance.

### Compare Two Weibull Distributions

Input the parameters of two Weibull distributions to compare their probability density functions, cumulative distribution functions, or other statistical properties. This allows you to assess differences and similarities between the distributions effectively.

### Calculate Weibull Entropy

Enter the shape and scale parameters to compute the entropy of the Weibull distribution, indicating the amount of uncertainty or randomness in the distribution. Understanding entropy is crucial in various fields like information theory and reliability analysis.

### Perform Sensitivity Analysis on Weibull Parameters

Vary the shape and scale parameters to observe how the outputs like probability density function or hazard function change. Sensitivity analysis helps you understand the impact of parameter changes on the distribution’s characteristics and informs better decision-making.