Hi,

I'm trying to evaluate the following integral:

$\int_{-\infty}^\infty \phi(x)\Phi(a+bx)^2dx$

where Phi is the cdf of a std Normal random variable, and phi is the pdf $(1\sqrt(2pi))exp(-x^2\2)$.

I have that $\int_{-\infty}^\infty \phi(x)\Phi(a+bx)dx = Phi(\frac{a}{\sqrt(1+b^2)})$, so I could integrate the above expression if I could evaluate

$\int_0^y \phi(x)\Phi(cx)dx$

By way of background, I have a discrete choice experiment where individual i has latent utility $u_{ij}$ for item $j$

$u_{ij}=X_{ij}(\beta+b_i) + \epsilon_{ij}$,

$\epsilon_{ij} \sim iid N(0,\sigma^2)$ $b_i \sim N(0,\eta^2)$ are random effects

and I am trying to express the intra-rater agreement as a function of $X$, $\beta$, $\sigma$ and $\eta$.

Any thoughts would be much appreciated!

Eleanor