Weibull Distribution Probability Calculator

Weibull Distribution Calculator

Calculate probabilities and values for the Weibull distribution.

Scale Parameter (λ):

Shape Parameter (k):

Select Calculation Type:

Score (X₁):

Probability (P(X ≤ x₁)):

Score (X₂):

Probability (P(X ≤ x₂)):

P(X ≤ x₁):

P(X > x₁):

Weibull Density (f(x₁)):

Score (x₁):

Score (x₂):

P(X ≤ x₂):

P(X > x₂):

Weibull Density (f(x₂)):

P(x₁ ≤ X ≤ x₂):

Understanding the Weibull Distribution

The Weibull distribution is a continuous probability distribution commonly used in reliability analysis, survival analysis, and risk assessment. It is highly flexible due to its two parameters, allowing it to model a variety of data types and patterns. The Weibull distribution can represent different life behaviors depending on its shape parameter \( k \), from failure rates increasing or decreasing over time to remaining constant.

Key Components of the Weibull Distribution

Scale Parameter (λ):
- λ (lambda): The scale of the distribution, which affects the spread of data. Higher values of \( \lambda \) stretch the distribution, while lower values compress it.
Shape Parameter (k):
- k: This parameter determines the shape of the distribution, influencing whether the failure rate increases, decreases, or remains constant over time. When \( k < 1 \), the failure rate decreases over time, typical of items that fail early. When \( k = 1 \), the Weibull distribution reduces to the Exponential distribution, with a constant failure rate. When \( k > 1 \), the failure rate increases over time, often modeling wear-out failures.
Probability (P): The probability that a random variable following the Weibull distribution is less than or equal to a given value \( X \).
Score (X): A specific value within the Weibull distribution used to calculate probabilities, representing time or life length in reliability contexts.

Weibull Distribution Formula

The probability density function (pdf) of the Weibull distribution with shape parameter \( k \) and scale parameter \( \lambda \) is given by:

\[ f(x; \lambda, k) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k - 1} e^{-(x / \lambda)^k} \quad \text{for } x \geq 0 \]

This formula gives the likelihood of observing each possible time \( x \) until failure. The cumulative distribution function (CDF), which gives the probability that \( X \) is less than or equal to \( x \), is:

\[ P(X \leq x) = 1 - e^{-(x / \lambda)^k} \quad \text{for } x \geq 0 \]

Conditions for Using the Weibull Distribution

Non-Negativity: The Weibull distribution only applies to positive values of \( X \), typically representing time-to-failure or lifetime data.
Shape Dependence: The suitability of the Weibull distribution depends on the shape parameter \( k \), which should align with the behavior of the data (e.g., increasing, decreasing, or constant failure rates).
Independence: Observations of time-to-failure should be independent for accurate modeling with the Weibull distribution.

Calculating Weibull Distribution Probabilities and Values

To compute Weibull distribution probabilities and values, we can use libraries in JavaScript, Python, and R. Here’s how to perform these calculations:

1. Using JavaScript

In JavaScript, you can use the jStat library for probability and inverse probability calculations.

// Calculate probability (CDF) for a given X
const lambda = 1.5; // Scale parameter
const k = 2.0; // Shape parameter
const x = 2.5; // Time until the event

// CDF: P(X ≤ x)
const cumulativeProbability = jStat.weibull.cdf(x, k, lambda);

// Inverse CDF: Get the time for a given probability (e.g., 0.95)
const probability = 0.95;
const timeForProbability = jStat.weibull.inv(probability, k, lambda);

console.log('Cumulative Probability:', cumulativeProbability);
console.log('Time for P(X ≤ x) = 0.95:', timeForProbability);

Note: Ensure you have the jStat library included in your project to perform these calculations.

2. Using Python

In Python, the SciPy library provides functions for cumulative probability and inverse calculations.

from scipy.stats import weibull_min

# Define the scale and shape parameters
lambda_ = 1.5  # Scale parameter
k = 2.0  # Shape parameter
x = 2.5  # Time until the event

# CDF: P(X ≤ x)
cumulative_probability = weibull_min.cdf(x, k, scale=lambda_)

# Inverse CDF: Get the time for a given probability (e.g., 0.95)
probability = 0.95
time_for_probability = weibull_min.ppf(probability, k, scale=lambda_)

print("Cumulative Probability:", cumulative_probability)
print("Time for P(X ≤ x) = 0.95:", time_for_probability)

Note: Install SciPy by running pip install scipy if it’s not already installed.

3. Using R

In R, the stats package includes functions to calculate Weibull distribution probabilities and values.

# Define the scale and shape parameters
lambda <- 1.5  # Scale parameter
k <- 2.0  # Shape parameter
x <- 2.5  # Time until the event

# CDF: P(X ≤ x)
cumulative_probability <- pweibull(x, shape=k, scale=lambda)

# Inverse CDF: Get the time for a given probability (e.g., 0.95)
probability <- 0.95
time_for_probability <- qweibull(probability, shape=k, scale=lambda)

cat("Cumulative Probability:", cumulative_probability, "\n")
cat("Time for P(X ≤ x) = 0.95:", time_for_probability, "\n")

Note: The pweibull function calculates the cumulative probability, while qweibull gives the time for a specified probability.