A very important distribution for analyzing the reliability of manufactured goods is the Weibull distribution, whose probability density function is defined by ƒ(x) = abx^b-1e^-axb for x in [0, ∞), where a and b are constants. Notice that when b = 1, this reduces to the exponential distribution. The Weibull distribution is more general than the exponential, because it applies even when the failure rate is not constant. Use Simpson’s rule with n = 100, or the integration feature on a graphing calculator, to approximate the following for the Weibull distribution with a = 4 and b = 1.5. Use a limit of 3 in place of ∞.

(a) The mean 

(b) The standard deviation