The life of a recirculating pump follows a Weibul...

The life of a recirculating pump follows a Weibul distribution with parameters $\beta=2$ and $\delta=700$ hours.
(a) Determine the mean life of a pump.
(b) Determine the variance of the life of a pump.
(c) What is the probability that a pump will last longer than its mean?

Key Concepts

Gamma Function

The Gamma function extends the concept of factorial to non-integer values and is integral in defining the moments of many continuous distributions, including the Weibull. It is frequently used in the formulas for calculating the mean and variance, as these often involve the Gamma function applied to functions of the distribution's parameters.

Mean and Moment Calculations

Calculating the mean (expected value) and variance of a distribution involves determining its moments—measures of central tendency and spread. For the Weibull distribution, these calculations depend on the Gamma function and the distribution's parameters, allowing analysts to quantify average performance and variability in lifetime or reliability analyses.

Survival Function

The survival function, which is one minus the cumulative distribution function (CDF), provides the probability that a component or system will operate beyond a given time. In the context of life data analysis, it is used to evaluate reliability or longevity, making it a key concept when assessing the performance of items modeled by distributions like the Weibull.

Weibull Distribution

The Weibull distribution is a continuous probability distribution widely used in reliability engineering and survival analysis. It is defined by a shape parameter and a scale parameter, which determine the failure rate behavior and distribution spread, making it useful for modeling life data and failure times.

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