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We define the following notions for a jointly normally distributed random vector $P=(P_1,...,P_n)$ with f the density function.

$$\mu=\int_{-\infty}^{\infty}(x_i-\mu_i)f_i(x_i)dx_i$$

$$\sigma^2_{ij}=\int_{-\infty}^{\infty}(x_i-\mu_i)(x_j-\mu_j)f_{ij}(x_i,x_j)dx_idx_j$$ where the density function of n-dimensional jointly normally distributed random vector is given by

$$f(x)=\frac{1}{(2\pi)^\frac{n}{2}\sqrt{\det{V}}}\exp\left(-\frac12 \left\langle x-\mu,V^{-1}(x-\mu)\right\rangle\right)$$

and where $V={(\sigma^2_{ij})}_{1\leq{i,j}\leq{n}}$ is the variance-covariance matrix.

  • I don't understand the density function of a jointly jointly normally distributed random vector.
  • To understand all of these above terms, is the knowledge of multi-variate normal distribution necessary?

If any quant-finance expert could explain me the above terms, it would be helpful to me in understanding quadratic finance.

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  • $\begingroup$ What is the meaning of $\langle{x-\mu,V^{-1}(x-\mu)\rangle}$ $\endgroup$ – Dhamnekar Winod Jun 6 '16 at 14:51
  • $\begingroup$ Short answer is yes, at least some basic knowledge of multivariate statistics would be necessary to understand the notion of the $n$-dimensional normal distribution. By the way, it seems to me there are some minor typos in your question, e.g. the $\mu_i$ on the rhs of the first equation. $\endgroup$ – Dr_Be Jun 6 '16 at 15:25

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