Determine the probability that a bearing lasts at least 6000 hours. Now, EXAMPLE 4.21 I Bearing Wear The time to failure (in hours) of a bearing in a mechanical shaft is satisfactorily modeled as a Weibull random variable with B = 1/2 and 8 = 5000 hours. Determine the mean time until failure. From the expression for the mean, 6000 P(X > 6000) = 1 – F(6000) = exp| 5000 = e-1.44 = 0.237 E(X) = 5000T[1+ (1/2)] = 500or[1.5] = 5000×0.5/T ni anoitonul vpienob vilidedong botoolaz lo adasn od nodw isi) so2 nso sw.nonomt ytiens Practical Interpretation: Consequently, only 23.7% of all ai nobudinalb lludisW orl = 4431.1 hours olludiialb dglolaH odt.oalA.gotudi bearings last at least 6000 hours.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...