Probability theory is the basis of mathematical statistics, which is an application of probability theory. The differences are as follows:

1. Different applications: Probability theory and mathematical statistics belong to a branch of mathematics, paying more attention to theoretical research, and their conclusions are widely used in the study of random phenomena in various fields.

Second, variables are different: social statistics describe variables, while mathematical statistics describe random variables.

Third, the form is different: statistics pays more attention to application, and many conclusions come from probability theory and mathematical statistics. Mathematical statistics pays more attention to the derivation of formulas, while statistical principles only transform the formulas of mathematical statistics into easier-to-use forms.

Fourth, the probability is different: probability studies the probability of a single event. Mathematical statistics study the sampling probability of a group. And the possible interval of this probability. Mathematical statistics is more inclined to the concept of statistics.

Characteristics of mathematical statistics

Based on the observation and experimental data of random phenomena, it studies random phenomena on the basis of probability theory, selects a mathematical model for random phenomena according to the data, verifies whether the mathematical model is suitable with mathematical data, and then studies its characteristics, properties and regularity on a suitable basis.

For example, the light bulb factory produces light bulbs, and several products are taken out for testing on a certain day. Before the test, we don't know the life, probability and distribution of light bulbs on that day. After the test, the service life of these bulbs is obtained as data, from which the service life and qualified rate of the whole batch of bulbs are inferred. In order to study its distribution, the mathematical model provided by probability theory is used to carry out exponential distribution, and the numerical value is obtained, and then the suitability of exponential distribution is determined through several days of sampling experiments.