# Differentiate the multivariate standard normal distribution

I have some questions relating to the multivariate cumulative distribution function.

Let $\mathcal{N}_n$ denotes the multivariate cumulative distribution function of Gaussian random variables, ie for $X_1$, $X_2$, ..., $X_n$ standard normal random variables $\mathcal{N}(0,1)$ with $\Sigma$ the correlation matrix of $X_i$, then for all $(x_1,...,x_n) \in \mathcal{R}^n, \mathcal{P}(X_1 \le x_1, ..., X_n \le x_n)=\mathcal{N}(x_1, ..., x_n, \Sigma)$.

For $n > 2$, I read $\mathcal{N}_n$ is given by

$$\mathcal{N}_n(t_1,.., t_n; \rho_1,.., \rho_{n-1}) = \int_{-\infty}^{t_1} ... \int_{-\infty}^{t_n} \frac{1}{(2 \pi)^{\frac{n}{2}} \prod_{i=1}^{n-1} \sqrt{1-\rho_i^2} } \exp \left( \frac{-x_1^2}{2} - \sum_{i=1}^{n-1} \frac{(x_{i+1}-\rho_i x_i)^2}{2(1-\rho_i)} \right) d x_1 ... d x_n$$

However, I do not know where it comes from and what really means these $\rho_i$. I would like to know if there exists formulae to compute the derivatives with respect to parameters of $t_i$, ie

$$\frac{\partial \mathcal{N}_n(t_1,.., t_n; \rho_1,.., \rho_{n-1})}{\partial c_1}$$

where $t_i = t_i(c_1,...,c_k)$.

I would start by writting

$$\frac{\partial \mathcal{N}_n(t_1,.., t_n; \rho_1,.., \rho_{n-1})}{\partial c_1} = \frac{\partial t_1}{\partial c_1} \times \int_{-\infty}^{t_2} ... \int_{-\infty}^{t_n} \frac{1}{(2 \pi)^{\frac{n}{2}} \prod_{i=1}^{n-1} \sqrt{1-\rho_i^2} } \exp \left( \frac{-t_1^2}{2} - \frac{(x_{2}-\rho_1 t_1)^2}{2(1-\rho_1)} - \sum_{i=2}^{n-1} \frac{(x_{i+1}-\rho_i x_i)^2}{2(1-\rho_i)} \right) d x_2 ... d x_n$$

• Can you provide the source of this formula for the normal cdf? – Xi'an Aug 31 '16 at 0:26
• Tristan Guillaume, Autocallable Structured Products p16, 2015. It seems coming from Tong Y.L. The multivariate normal distribution. Springer but I did not fond thé formula – Stephane Aug 31 '16 at 9:01