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$$\normalsize\ percentile\ x (x≧0)= 1$$ $$\normalsize\ shape\ parameter\ a (a＞0)= 2$$ $$\normalsize\ scale\ parameter\ b(b＞0)= 1$$

$$probability\ density\ f\ =0.7357588823428846431911$$ $$lower\ cumulative\ P\ =0.6321205588285576784045$$

$$\normalsize Weibull\ distribution$$ $$(1)\ probability\ density\\ \hspace{30px}f(x,a,b)={\large\frac{a}{b}(\frac{x}{b})^{a-1}e^{-(\frac{x}{b})^a}}$$ $$(2)\ lower\ cumulative\ distribution$$ $$ \hspace{30px}P(x,a,b)={\large\int_{\small 0}^{\small x}}f(t,a,b)dt={\large 1-e^{-(\frac{x}{b})^a}}$$

The Wei-bull Distribution is a persistent likelihood appropriation named after Swedish mathematician Waloddi Wei-bull. He initially proposed the dispersion as a model for material breaking strength, however perceived the capability of the circulation in his 1951 paper A Statistical Distribution Function of Wide Applicability. Today, it's generally used to survey item unwavering quality, investigate life information, and model failure times. The Wei-bull can likewise fit a wide scope of information from numerous different fields, including science, financial aspects, designing sciences, and hydrology (Rinne, 2008).

In spite of the fact that it's incredibly valuable as a rule, the Wei-bull is definitely not a suitable model for each circumstance. For instance, compound responses and erosion Distribution are generally demonstrated with the log-normal conveyance.

In the event that x speaks to "time-to-disappointment", the Weibull dispersion is portrayed by the way that the disappointment rate is corresponding to an intensity of time, specifically β – 1. Accordingly β can be deciphered as follows:

- An estimation of β < 1 shows that the disappointment rate diminishes over the long run. This occurs if there is critical "newborn child mortality", or where inadequate things bomb ahead of schedule with a disappointment rate diminishing over the long haul as the faulty things are removed of the populace.
- An estimation of β = 1 demonstrates that the disappointment rate is consistent after some time. This may propose arbitrary outer occasions are causing mortality or failure.
- An estimation of β > 1 shows that the disappointment rate increments with time. This occurs if there is an "maturing" measure; for example on the off chance that parts are bound to wear out as well as come up short over the long haul.
- 1/α can be seen as the failure rate. The mean of the Wei-bull distribution is the interim to failure (MTTF) or interim between disappointments (MTBF) = \alpha \Gamma \! \left( \! 1+\frac{1}{\beta} \! \right).

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