Weibullpylväsd

Weibullpylväsd, often referred to as the Weibull distribution, is a continuous probability distribution. It is named after the Swedish mathematician Waloddi Weibull. The distribution is characterized by its ability to model a wide range of failure rates and is therefore widely used in reliability engineering and survival analysis.

The probability density function (PDF) of the Weibull distribution is given by f(x; k, lambda) = (k/lambda) * (x/lambda)^(k-1) * exp(-(x/lambda)^k) for x >= 0. Here, k is the shape parameter and lambda is the scale parameter. The shape parameter k determines the nature of the failure rate. If k < 1, the failure rate decreases over time, suggesting infant mortality. If k = 1, the failure rate is constant, similar to an exponential distribution. If k > 1, the failure rate increases over time, indicating wear-out. The scale parameter lambda influences the spread or location of the distribution.

The cumulative distribution function (CDF) is F(x; k, lambda) = 1 - exp(-(x/lambda)^k). This function gives the probability that the random variable X is less than or equal to a certain value x. The Weibull distribution is particularly useful because it can approximate many other distributions by adjusting its parameters. Its flexibility makes it a powerful tool for analyzing the lifetime of components, systems, and even the duration of biological processes.