# Evaluating $\mathbb{E} \left[ \Phi\left(\frac{X}{c}\right) \Phi\left(\frac{Y}{c} \right)\right]$

I'm looking for a way to evaluate the expectation

$$\mathbb{E}\left[ \Phi\left(\frac{X}{c}\right) \Phi\left(\frac{Y}{c}\right) \right]$$

where $\Phi(x)$ denotes the cdf of a standard normal distribution and $X$ and $Y$ follow a standard bivariate normal distribution, i.e. a bivariate normal distribution with means equal to zero, variances equal to 1 and correlation coefficient equal to $\rho$.

I have found this proof but I'm having some trouble understanding it so could someone please explain to me how we get from the first equality to the second? I can take over from there.

• you mean $(X,Y) \sim \mathcal{N}(\mu,\Sigma)$ follows a $2$-dimensional normal distribution with mean $\mu = (0,0)$ and covariance matrix $\Sigma = \left( \begin{array}{ll} \sigma^2 & \rho \\ \rho & \sigma^2 \end{array}\right)$ (X,Y) \sim \mathcal{N}(\mu,\Sigma), \Sigma = \left( \ begin{array}{ll} \sigma^2 & \rho \\ \rho & \sigma^2 \end{array}\right) – reuns Mar 16 '16 at 20:05

• $E1_{A\cap B}=E1_{A}1_{B}=E1_{A}E1_{B}$. First equality stems from properties of indicator functions, and the second from independence of $Z_1$ and $Z_2$. – V. Vancak Mar 16 '16 at 20:38