# Derivatives of the multivariate normal CDF w.r.t. its parameters

Let $\boldsymbol X \sim \mathcal N_d(\boldsymbol\mu, \boldsymbol\Sigma)$ and let $f_{\boldsymbol X}(\boldsymbol x)$ be its PDF. The CDF is $$F_{\boldsymbol X}(\boldsymbol q) = \int_{-\infty}^{q_1} \dots \int_{-\infty}^{q_d} f_{\boldsymbol X}(\boldsymbol x)\ dx_d \dots dx_1.$$ How do I compute $\partial F_{\boldsymbol X}(\boldsymbol q) / \partial\boldsymbol\mu$ and $\partial F_{\boldsymbol X}(\boldsymbol q) / \partial\boldsymbol\Sigma$? I know the derivatives of the PDF, but I am unsure about the integrals. Can I apply some chain rule? Any hints appreciated.

• Under certain conditions (for example, Fubini's theorem), you can interchange the order of integration and differentiation. That should get you started... – jbowman Mar 10 '18 at 22:50
• If I wanted to differentiate w.r.t. $\boldsymbol x$, Fubini would certainly help, but how can I use Fubini with $\boldsymbol\mu$ or $\boldsymbol\Sigma$? Could you say a few more words, @jbowman? – bbrot Mar 11 '18 at 0:13
• Well, less indirectly, it's a special case of Leibniz' rule, and Fubini is used in a proof of it. Sorry about the imprecision. The point is that you can just interchange the order of integration and differentiation. – jbowman Mar 11 '18 at 0:18