Weibull Distribution Calculator

Compute the PDF, CDF (lower probability), reliability (survival function), mean, and variance of a Weibull distribution with shape k and scale lambda. Built for reliability engineering and life-data analysis.

Input

Enter the shape parameter k, the scale parameter lambda, and a value x to compute the Weibull probability density, cumulative distribution (lower probability), reliability, mean, and variance.

Value x to evaluate (x ≥ 0)

A non-negative value such as a time or lifetime at which to evaluate the probability.

Shape parameter k

A positive value setting the failure-rate trend: below 1 early failures, 1 constant, above 1 wear-out.

Scale parameter lambda

A positive value for the characteristic life, the time by which about 63.2 percent have failed.

Result

Cumulative distribution (lower probability) for x = 1,000, k = 1.5, lambda = 1,200

0.53267287

Reliability R(x) (survival)

0.46732713

Probability density f(x)

0.00053326

Mean

1,083.29435154

Variance

540,994.01013206

Probability density function PDF

Cumulative distribution function CDF

How it works

The Weibull distribution is a continuous probability distribution with a shape parameter k and a scale parameter lambda, widely used in reliability engineering to model product lifetimes and failure times.
The probability density function is f(x) = (k/lambda)(x/lambda)^(k−1) e^(−(x/lambda)^k), and the cumulative distribution function is F(x) = 1 − e^(−(x/lambda)^k) for x ≥ 0.
The upper-tail probability R(x) = 1 − F(x) = e^(−(x/lambda)^k) is called the reliability or survival function and gives the probability of operating without failure up to time x.
When the shape parameter k is below 1 the distribution models early-life failures (decreasing failure rate); k = 1 reduces to the exponential distribution (constant rate); k above 1 models wear-out failures (increasing rate).
The mean is lambda·Gamma(1 + 1/k) and the variance is lambda²[Gamma(1 + 2/k) − Gamma(1 + 1/k)²]. Gamma is the gamma function, evaluated here with a Lanczos approximation.
These results are approximate and should be verified by a qualified specialist before use in any final reliability assessment or warranty design decision.

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