Standard Normal Distribution Density Function
This section demostrates how to generate bivariate normal distribution density function for both "with correlation" and "without correlation". A bivariate normal distribution without correlation (means X and Y are independent) is simply the product of the two normal distributions. For example:
In the case of the dependency (correlation is not zero), the density function is as followed:
Note that this formula is good whether there is a correlation or not.
In the example below, given the standard deviation and mean for the two variables, 1 and 0 respectively, and the correlation of 0, we derive the following probability distribution table:
Here are the definition of the variables in the formula above:
C15 = standard deviation of x
C16 = mean of x
G15 = standard deviation of y
G16 = mean of y
F21 = correlation of x and y
J23 = x
I44 = y
Below are the charts for the above senerio (no correlation with standard = 1, and mean = 0). The 3 charts are taken from different angles to help users gain a better prospective on this distribution.
Here are what the charts look like when the correlation is 0.7 (note that correlation cannot equals to 1, else the formula will throw an error because the square root of 0 is undefinite). As the correlation becomes larger, the distribution tends to be narrower.