Weibull Distribution 

The Weibull distribution is one of the most commonly used distributions in
reliability. It is commonly used to model time to fail, time to repair and material
strength. The Weibull probability density function is
where b is the shape parameter, The shape parameter is what gives the Weibull distribution its flexibility. By changing the value of the shape parameter, the Weibull distribution can model a wide variety of data. If b = 1 the Weibull distribution is identical to the exponential distribution, if b = 2, the Weibull distribution is identical to the Rayleigh distribution; if b is between 3 and 4 the Weibull distribution approximates the normal distribution. The Weibull distribution approximates the lognormal distribution for several values of b. For most populations more that fifty samples are required to differentiate between the Weibull and lognormal distributions. A sample of the Weibull distribution's flexibility is shown in the figure below with q = 100 and d = 0. The scale parameter determines the range of the distribution. The scale parameter is also known as the characteristic life if the location parameter is equal to zero. If d does not equal zero, the characteristic life is equal to q+d; 63.2% of all values fall below the characteristic life regardless of the value of the shape parameter. The effect of the scale parameter of the probability density function is shown in the figure below. The location parameter is used to define a failurefree zone. The probability of failure when x is less than d is zero. When d>0, there is a period when no failures can occur. When d<0, failures have occurred before time equals 0. At first this seems ridiculous, but a negative location parameter is caused by shipping failed units, failures during transportation, and shelf life failures. Generally, the location parameter is assumed to be zero. The effect of the location parameter is shown in the figure below. The Weibull hazard function is determined by the value of the shape parameter. When b<1 the hazard function is decreasing; this is known as the infant mortality period. When b=1, the failure rate is constant. When b>1 the failure rate is increasing; this is known as the wearout period. The Weibull hazard function is shown in the figure below. The Weibull reliability and cumulative distribution functions are Example: The time to fail for a flexible membrane follows the Weibull distribution with b = 2 and q = 300 months. What is the reliability at 200 months? After how many months is 90% reliability achieved? Solution: After 200 months, the reliability of the flexible membrane is By manipulating the expression for reliability, 90% reliability is achieved after Click Here to download this solution in a Microsoft Excel spreadsheet.
The mean and variance of the Weibull distribution are computed using the gamma distribution which is available in Microsoft Excel or Lotus 123. The mean of the Weibull distribution is The mean of the Weibull distribution is equal to the characteristic life if the shape parameter is equal to one. The mean as a function of the shape parameter is shown in the figure below. The variance of the Weibull distribution is The variance of the Weibull distribution decreases as the value of the shape parameter increases. This is shown in the figure below. The reliability measure of many components is the mean time to fail. Consumer electronics companies often advertise the mean time to fail of the products. The mean time to fail is a deceptive measure because the variance of the time to fail distribution is not considered. To achieve the same reliability with a larger variance, requires a larger mean. Consider two components; A and B. Component A has a mean time to fail of 4645 hours, and component B has a mean time to fail of 300 hours. If both components sell for the same price, which component should be used to maximize reliability at 100 hours? This question cannot be answered without knowing more information about the distribution of the time to fail. Component A has a mean of 4645 hours and a Weibull time to fail distribution with a shape parameter of 0.8. Using the mean and the shape parameter, the scale parameter of component A can be computed to be 4100 hours. The reliability at 100 hours is Component B has a mean of 300 hours and a Weibull time to fail distribution with a shape parameter of 3. Using the mean and the shape parameter, the scale parameter of component B can be computed to be 336 hours. The reliability at 100 hours is Although the mean of component A is more than 10 times as large as the mean of component B, the reliability of component B is greater than the reliability of component A at 100 hours. Continuing with this example, if the reliability at 1000 hours is to be maximized, component A has a reliability of 0.723 and component B has a reliability of approximately zero. 