The article “Stochastic Modeling for Pavement Warranty Cost Estimation” (J. of Constr. E...

The article “Stochastic Modeling for Pavement Warranty Cost Estimation” (J. of Constr. Engr. and Mgmnt., 2009: 352–359) proposes the following model for the distribution of Y = time to pavement failure. Let X₁ be the time to failure due to rutting, and X₂ be the time to failure due to transverse cracking; these two rvs are assumed independent. Then Y = min(X₁, X₂). The probability of failure due to either one of these distress modes is assumed to be an increasing function of time t. After making certain distributional assumptions, the following form of the cdf for each mode is obtained:

where Φ is the standard normal cdf. Values of the five parameters a, b, c, d, and e are –25.49, 1.15, 4.45, –1.78, and .171 for cracking and –21.27, .0325, .972,–.00028, and .00022 for rutting. Determine the probability of pavement failure within t = 5 years and also t = 10 years.