Supplementary mathematics/Law of total probability

The law of total probability, which is also known as Bayes' law of probability, is a rule of law in the field of statistics and probability that divides probability calculations into a separate, detailed and advanced part, that is, for example, you have no relative knowledge of the probability of the occurrence of an expression B, and for Finding the probability of an event B that we know, we can use to find the probability of the event B.

Multivariate Law of Total Probability
According to this pattern B1,..., Bn and an event called set A, the probability of event A can be calculated as a quantitative or weighted average in which the pattern B1,..., Bn plays a role to find the calculation of its occurrence. According to each occurrence in the partition pattern with the probability of the occurrence of partitioning in terms of quantity, this formula can be used:

$$ \Rho (\Alpha)=\Rho (\Alpha|\Beta _{1 })\Rho(\Beta _{1 })+.....+\Rho (\Alpha|\Beta _{n})\Rho(\Beta _{n})$$

The same idea can be applied and calculated for random vectors. In this way, it is possible to obtain a

For the marginal distribution, we pooled other variables from the joint distribution:

$$f (\Chi)\Chi _{1 }= \int\limits _{x_2 }^{ }  ...\int f \Chi_1 \Chi_2 (x_1, x_2)dx_2$$

The law of total probability is both an independent event and the probability of evil, and it is a general law for these two topics. From the definition of conditional probability, we know that