EX 4.28

Project managers often use a method labeled PERT-for program evaluation and review technique-to coordinate the various activities making up a large project. (One successful application was in the construction of the Apollo spacecraft.) A standard assumption in PERT analysis is that the time necessary to complete any particular activity once it has been started has a beta distribution with $A=$ the optimistic time (if everything goes well) and $B=$ the pessimistic time (if everything goes badly). Suppose that in constructing a single-family house, the time $X$ (in days) necessary for laying the foundation has a beta distribution with $A=2,B=5,\alpha =2$ , and $\beta =3$ . Then $\alpha /\left(\alpha +\beta \right)=.4$ , so $E\left(X\right)=2+\left(3\right)\left(.4\right)=3.2$ . For these values of $\alpha$ and $\beta$ , the pdf of $X$ is a simple polynomial function. The probability that it takes at most 3 days to lay the foundation is

$$P\left(X\le 3\right)={\int }_2^3\frac{1}{3}\cdot \frac{4!}{1!2!}\left(\frac{x-2}{3}\right){\left(\frac{5-x}{3}\right)}^{2}dx$$ $$=\frac{4}{27}{\int }_2^3\left(x-2\right){\left(5-x\right)}^{2}dx=\frac{4}{27}\cdot \frac{11}{4}=\frac{11}{27}=.407$$