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The Weibull distribution is a probability distribution used to model the time-to-failure of a system.

The Weibull distribution is named after Swedish mathematician Waloddi Weibull. It is a continuous probability distribution that is widely used in reliability engineering to model the time-to-failure of a system. The distribution is characterized by two parameters: the shape parameter (k) and the scale parameter (λ). The shape parameter determines the shape of the distribution curve, while the scale parameter determines the location of the curve along the x-axis.

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

f(x) = (k/λ) * (x/λ)^(k-1) * e^(-(x/λ)^k)

where x is the time-to-failure, k is the shape parameter, and λ is the scale parameter.

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

F(x) = 1 - e^(-(x/λ)^k)

The mean and variance of the Weibull distribution are given by:

μ = λ * Γ(1 + 1/k)

σ^2 = λ^2 * (Γ(1 + 2/k) - Γ^2(1 + 1/k))

where Γ is the gamma function.

The Weibull distribution is commonly used in reliability engineering to model the time-to-failure of a system. It is also used in other fields such as finance, economics, and meteorology.

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