A calculator for the Bivariate Normal
At the bottom of the page at the link, under "A calculator for cumulative probabilities from the bivariate normal distribution" there's a little binary applet, which can be downloaded, and which calculates the percentage chances of being in various parts of the volume of the distribution.
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Yes. And that is true of most probability distributions.
Yes. Most do.
It is a continuous parametric distribution belonging to the family of exponential distributions. It is also symmetric.
Because very many variables tend to have the Gaussian distribution. Furthermore, even if the underlying distribution is non-Gaussian, the distribution of the means of repeated samples will be Gaussian. As a result, the Gaussian distributions are also referred to as Normal.
You calculate the z-scores and then use published tables.