EX 4.26

An understanding of the volumetric properties of asphalt is important in designing mixtures which will result in high-durability pavement. The article “Is a Normal Distribution the Most Appropriate Statistical Distribution for Volumetric Properties in Asphalt Mixtures?” (J. of Testing and Evaluation, Sept. 2009: 1-11) used the analysis of some sample data to recommend that for a particular mixture, $X=$ air void volume (%) be modeled with a three-parameter Weibull distribution. Suppose the values of the parameters are $\gamma =4,\alpha =1.3$ , and $\beta =.8$ (quite close to estimates given in the article).

For $x>4$ , the cumulative distribution function is

$$F\left(x;\alpha ,\beta ,\gamma \right)=F\left(x;1.3,.8,4\right)=1-e^{-{\left[\left(x-4\right)/.8\right]}^{1.3}}$$

The probability that the air void volume of a specimen is between $5\%$ and $6\%$ is

$$P\left(5\le X\le 6\right)=F\left(6;1.3,.8,4\right)-F\left(5;1.3,.8,4\right)=e^{-{\left[\left(5-4\right)/.8\right]}^{1.3}}-e^{-{\left[\left(6-4\right)/.8\right]}^{1.3}}$$ $$=.263-.037=.226$$

Figure 4.29 shows a graph from Minitab of the corresponding Weibull density function in which the shaded area corresponds to the probability just calculated.

Figure 4.29 Weibull density curve with threshold $=4$ , shape $=1.3$ , scale $=.8$