Multivariate Standard Normal Probability Distribution
This example is a more advanced version of the Monte Carlo Integration example given earlier. In addition to the material taken from the example mentioned above, this program also utilized a numerical procedure (specifically, Jocobi search method, for derivation of the Eigenvectors and Eigenvalues) and matrix algebra. The procedure for generating random numbers from a multivariate distribution is described in the 4 steps of the example shown later.
This program computes probability from a multivariate standard normal probability distribution given the z values and the correlations for up to 5 variables. The user can increase the number of variables by a slight modification of the program. A graphical representation of a bivariate standard normal distribution with no dependency between the two variables is shown in Figure 1.

In order for the program to work properly, the user need to place the input value in the same cell references as in Figure 2 or 3. Cell E14 of Figure 2 shows the probability from a bivariate standard normal distribution given z1 = 1 and z2 = 0 with correlation of 0.4. The probability in this case is 45.79% based on 100,000 iterations. Note that, unlike the probability derive from the Monte Carlo Integration program, which is computed based on the area under standard normal probability distribution from 0 to z. The probability computed in this program is based on the area under the probability distribution from negative infinity to z. Figure 3 shows the output when 5 variables are used.

The area of the cells, A16:F29, displays the standard deviation, the mean, and the correlation matrix for each variable based on the generated random numbers. The purpose of this section is to check how well the random numbers meet the specification. For example, the standard deviation and the mean from a standard normal probability distribution should be 1 and 0, respectively. The correlation matrix from the output should also match closely with the input.
Example of derivation of bivariate standard normal random numbers:
Figure 1. Bivariate Standard Normal Plot With Independency - Correlatoion = 0
Figure 2. Bivariate Standard Normal Computation

Figure 3. Multivariate Standard Normal Computation With 5 Variables

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