# How to scale (in the range 0 to 1) and mathematically explain two mutually exclusive probabilities of a data point belonging to a normal distribution?

I have two set of n-dimensional multivariate data, with the assumption that both set of data is normally distributed. When I get a new data point, my goal is to classify it into one of the two sets. The sum of probability of my new data point belonging to both the distribution therefore should be 1. I am using this to calculate the probability of the new data point belonging to each of the distribution. However, I am getting probability scores in the range 0 to 1 for both the cases resulting in the sum of probability being sometimes greater than 1 and sometimes less than 1. I don't know much statistics, is there any way to combine probability in the range 0 to 1, and explain it mathematically, or is there some other method to get classify new data into one of the two distributions, which satisfies the probability law of sum being equal to 1? Thanks in advance.

If your original observation is not extreme in either of the two distributions then it is possible to get the situation you describe, with the two calculations adding up to more than $$1$$. If your original observation is extreme in both distributions you can get the two calculations adding up to less than $$1$$. There is not reason to expect the two calculations to add to $$1$$.
One approach could be to compare the likelihoods of your observation under the two distributions (for continuous distributions proportional to the densities at that point, while your earlier calculations were related to the cumulative distribution functions) and then have probabilities calculated as $$\frac{f_1(x)}{f_1(x)+f_2(x)} \text{ and }\frac{f_2(x)}{f_1(x)+f_2(x)}$$ which will then clearly meet the conditions you have set. You would get the same result using Bayes' theorem if you had prior probabilities for the two original distributions of $$\frac12$$ each.
• The density for a multivatiate normal is $(2\pi)^{-\frac{k}{2}}\det(\boldsymbol\Sigma)^{-\frac{1}{2}} \, e^{ -\frac{1}{2}(\mathbf{x} - \boldsymbol\mu)^{{{\!\mathsf{T}}}} \boldsymbol\Sigma^{-1}(\mathbf{x} - \boldsymbol\mu) }$ May 21, 2021 at 10:59