For one of the engineering applications, we are required to calculate the integral of the product of a lognormal PDF and normal CDF. Essentially, the probablity of event occurance has a lognormal distribution, but once the event occurs, the probability of it to fail is normal. Hence, to compute the overall probability it is required to compute:
$\int \phi(x)\Phi(z) dx$,
where $\phi(x)$ = normal probability density function
$\Phi(z)$ = normal cumulative distribution function of z, where z is defined as:
$$
z = z_0 + a_0e^{bx}
$$
Any suggestions or thoughts would be greatly helpful.
Thanks.
