Statistics and probability - Weibull distribution and failure rate  

• Statistics distributions
Statistics and probability - Weibull distribution and failure rate.
The Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events.
If we let β equals 1, the Weibull distribution reduces to the exponential distribution. For values of β larger than 1, the curves become somewhat bell shaped and resemble the normal curve but display some skewness.
Like the gamma and exponential distributions, the Weibull distribution is also applied to reliability and life-testing problems such as the time to failure or life length of a component, measured from some specified time until it fails.
Let us represent this time to failure by the continuous random variable T, with probability density function f(t), where f(t) is the Weibull distribution. The Weibull distribution has inherent flexibility in that it does not require the lack of memory property of the exponential distribution. The cumulative distribution function (cdf) for the Weibull can be written in closed form and certainly is useful in computing probabilities.
When the Weibull distribution applies, it is often helpful to determine the failure rate (sometimes called the hazard rate) in order to get a sense of wear or deterioration of the component. Let us first define the reliability of a component or product as the probability that it will function properly for at least a specified time under specified experimental conditions.
The quantity Z(t) is aptly named as a failure rate since it does quantify the rate of change over time of the conditional probability that the component lasts an additional dt given that it has lasted to time t. The rate of decrease (or increase) with time is important. The following are crucial points.
(a) If β = 1, the failure rate = α, a constant. This, as indicated earlier, is the special case of the exponential distribution in which lack of memory prevails.
(b) If β larger than 1, Z(t) is an increasing function of time t, which indicates that the component wears over time.
(c) If β less than 1, Z(t) is a decreasing function of time t and hence the component strengthens or hardens over time.
#statistics
#probability
#distribution
#weibull
#weibulldistribution
#failurerate
#statistics32