# Weibull Distribution Calculator

The Weibull distribution is a two-parameter probability density function used in predicting the time to failure. It is often applied in manufacturing and materials science. The probability density function and cumulative distribution function are

pdf(x) = αβ^{-α}x^{α-1}e^{-(x/β)α} [0, ∞)

CDF(x) = 1 - e^{-(x/β)α} [0, ∞)

where α and β are the two parameters, both of which must be greater than 0. The parameter α is called the *shape parameter* because it determines the basic shape of the function; β is called the *scaling parameter* because it governs the horizontal stretching of the graph. When α = 1, the Weibull distribution becomes the standard exponential distribution

g(x) = (1/β)e^{-x/β},

and when α = 2, the Weibull distribution becomes the Rayleigh distribution

h(x) = (2x/β^{2})e^{-x2/β2}.

Thus, the Weibull distribution is a more general probability density function that includes other functions as special cases.

You can compute probability, mean, variance, standard deviation, mode, median, and Shannon's entropy (information entropy) using the formulas below, or by plugging the parameters into the calculator above. It will also calculate the probability that a random variable X is between X_{1} and X_{2}.

#### Computing P(X_{1} < X < X_{2})

Plug the values of X_{1}and X

_{2}into the CDF, then subtract. Thus, if X is a random Weibull-distributed variable, then

P(X

_{1}< X < X

_{2}) = e

^{-(X1/β)α}- e

^{-(X2/β)α}

#### Computing the Mean

The mean of a continuous probability distribution p(x) is found by evaluating the integral ∫xp(x) dx over its domain. In the case of the Weibull distribution, the mean isμ = βΓ(1 + 1/α),

where Γ is the Gamma Function.

#### Computing the Variance and Standard Deviation

The variance of a continuous probability distribution is found by computing the integral ∫(x-μ)²p(x) dx over its domain. For the Weibull distribution, the variance isσ² = β²[Γ(1 + 2/α) - Γ(1 + 1/α)²].

The standard deviation σ is the square root of the variance.

#### Computing the Median

The median of a continuous distribution function is a number*m*such that the integral ∫

^{m}

_{o}p(x) dx = 1/2. For the Weibull probability density function,

m = β(LN(2))

^{1/α}.

#### Computing the Mode

The mode of probability distribution is the most frequently occurring value. There may be more than one mode in some distributions and random samples. In a continuous distribution, the mode is the max of the function. The Weibull distribution's mode is given by the equationmode = β(1 - 1/α)

^{1/α}.

#### Computing the Entropy

To find the entropy of a continuous probability distribution, you calculate the integral ∫p(x)LN(p(x)) dx over the function's domain. In the case of a Weibull distribution, the entropy is given by the formulaentropy = 1 + γ(1 - 1/α) + LN(β/α),

where γ is the Euler-Mascheroni constant 0.57721566490153286060651209... In statistics, information theory, and probability theory, entropy is a measure of how unpredictable a system is.

© *Had2Know 2010
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