Calculate Weibull distribution probabilities and quantiles by entering the shape parameter (α), scale parameter (β), and either an x value or a probability p. Choose between P(X < x) (left-tail) and P(X > x) (right-tail) to get the corresponding probability or percentile.
Probability P(X < x) or P(X > x)
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Quantile x (from p)
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PDF f(x)
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Distribution Mean
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Standard Deviation
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The Weibull distribution is widely used in reliability engineering and survival analysis to model the time until failure of a component or system. It is flexible enough to represent increasing, decreasing, or constant failure rates depending on the shape parameter α.
The shape parameter α determines the behavior of the failure rate: α < 1 means decreasing failure rate (early failures), α = 1 reduces the Weibull to an exponential distribution (constant failure rate), and α > 1 means increasing failure rate (wear-out failures). The scale parameter β stretches or compresses the distribution along the x-axis and represents a characteristic life value.
Select 'P(X < x) — Left-tail' from the Probability Type dropdown, enter your shape α, scale β, and the x value. The calculator uses the Weibull CDF formula F(x) = 1 − exp(−(x/β)^α) to compute the probability that the random variable is less than x.
Enter a probability value p between 0 and 1 in the 'Probability (p)' field while leaving x blank (or it will be overridden). The calculator inverts the CDF to find x such that P(X < x) = p, using the formula x = β × (−ln(1 − p))^(1/α).
The probability density function is f(x) = (α/β) × (x/β)^(α−1) × exp(−(x/β)^α) for x > 0. This gives the density at a specific point x and is used to build the distribution curve shown in the chart.
The mean of a Weibull(α, β) distribution is β × Γ(1 + 1/α), where Γ is the gamma function. This calculator approximates the gamma function and displays the mean along with the standard deviation for your chosen parameters.
Yes. When α = 1, the Weibull distribution is identical to the exponential distribution with rate 1/β. When α = 2, it becomes the Rayleigh distribution. This flexibility makes it one of the most versatile continuous distributions in statistics and engineering.
P(X < x) is the left-tail (cumulative) probability — the area under the PDF curve to the left of x. P(X > x) is the right-tail (survival) probability — the area to the right, equal to 1 − P(X < x). Right-tail probabilities are often used in reliability contexts to express the probability that a component survives beyond time x.