I understand that there is no closed form expression for the general Normal Distribution function

But what if we restrict it to the simplest case?

Suppose we have jointly bivariate normal random variables $(X,Y)$ such that the following is true:

$E(X,Y) = (0,0)$

$Var(X,Y) = \begin{bmatrix}1 & \rho \\ \rho & 1\end{bmatrix}$

Then this simplifies the density function given here significantly. So finding a closed form expression is just solving the integral $\int_{-\infty}^x \int_{-\infty}^y f(u,v) du dv$.

So my question is, under the above conditions is it possible to find a solution to this integral?

  • 2
    $\begingroup$ Closed form is what you said first. A solution can always be found but not in closed form even with the simplifications you've made. $\endgroup$ Jan 22, 2017 at 19:32
  • 3
    $\begingroup$ By standardizing the distribution you just performed a linear transformation of the variables. If a closed form existed for the simplified case then you could just convert it to the general case with the reverse linear transformation $\endgroup$
    – Hugh
    Jan 22, 2017 at 19:34
  • $\begingroup$ Oh right that makes sense @Hugh. I ask because I am trying to perform differential evolution in R using DEoptim and for some reason the functions in packages "mvtnorm" and "pbivnorm" cause DEoptim to get stuck on the first iteration. It is unrelated to the question, but do either of you have any suggestions for alternative ways I can express bivariate normal distribution function without these packages? $\endgroup$
    – Patty
    Jan 22, 2017 at 19:46
  • $\begingroup$ You could compute the integral numerically. I don't know if that is what you had in mind. $\endgroup$ Jan 22, 2017 at 22:19


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