// Home / Documentation / Gallery of Distributions / Normal distribution (Multi variables)

# Normal distribution (Multi variables)

## Where do you meet this distribution?

• Finance, Economics : Value at Risk

## Shape of Distribution

### Basic Properties

• For n-variables, means for each variable (n-dimensional vector), standard deviations for each variable (n-dimensional vector $\mu$) and correlation coefficients for every pair of variables (n times n matrix. Called correlation matrix) are required (In practice, a covariance matrix $\Sigma$ calculated from standard deviation vector and correlation correlation matrix is required).
• Continuous distribution defined on $x\in R^{n}$

### Probability

• NTBINROMDIST is singular, when correlation coefficient $\rho=\pm 1$ and x1=x2.
• Probability density function
$f(x)=\frac{1}{(2\pi)^{n/2}\left|\Sigma\right|^{1/2}}\exp\left[-\frac{1}{2}(x-\mu)^\prime\Sigma^{-1}(x-\mu)\right]$
• li>How to compute these on Excel.

1
2

3

4
5
6
7
8
9
10

11

A B
Data Description
1.5 Value of 1st. variable for which you want the distribution
-1 Value of 2nd. variable for which you want the distribution
1.5 Mean of variable1 M1
-1.2 Mean of variabel2 M2
2 Standard deviation of variable1 Sigma1
0.7 Standard deviation of variabel2 Sigma2
0.5 Correlation coefficient Rho
Formula Description (Result)
=NTBINORMDIST((A2-A4)/A6,(A3-A5)/A7,A8,TRUE) Cumulative distribution function for the terms above
=NTBINORMDIST((A2-A4)/A6,(A3-A5)/A7,A8,FALSE) Probability density function for the terms above

• Function reference : NTBINORMDIST

## Random Numbers

• How to generate random numbers on Excel.

1
2
3
4
5
6
7
8

A B C D
Data Data Data Description
1.44 0.48 -0.36 cov. matrix
0.48 0.64 0 cov. matrix
-0.36 0 0.25 cov. matrix
Data Data Data Description
1 2 3 mean vector
Formula     Description (Result)
=NTRANDMULTINORM(100,A2:C4,A6:C6)     100 normal deviates (x 3 variables) for the terms above