How is the probability distribution of a normal distribution arranged in binary classification?

In binary classification, the normal distribution is often used to model the probabilities of objects belonging to classes, especially in probabilistic models such as naive Bayes classifiers or linear models with assumptions of normality of errors.

If we assume that data for each class are normally (Gaussian) distributed with different means and variances, then the probability of an object belonging to each class can be calculated using the normal distribution density formula:

[ p(x|class) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} ]

Where (\mu) and (\sigma^2) are the mean and variance of the feature for the given class.

Then, using Bayes' theorem, the posterior probability of the class is calculated:

[ p(class|x) = \frac{p(x|class) p(class)}{p(x)} ]

Where (p(class)) is the prior probability of the class.

Thus, the normal distribution helps estimate how likely it is that an object belongs to a particular class, which is used for decision-making in binary classification.