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Probability Theory and Mathematical Statistics Note

Probability Theory and Mathematical Statistics is an interesting subject, compared with other mathematics courses, it is not so tough. It is widely used in various aspects, include machine learning, information theory, financial mathematics, etc. I reference a sentence from Wiki to describe this subject:

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms.

The overall structure of probability theory can be described as following:

  1. Basic knowledge, include the definition of probability space, expectation, and variance;
  2. Various probability distributions include the discrete distributions and continuous distributions. For example:
    • Binomial distribution
    • Poisson distribution
    • Hypergeometric distribution
    • Geometric distribution
    • Normal distribution
    • Index distribution
    • Gamma distribution
    • Beta distribution
  3. Description of several feature parameters, which play an important role in analyzing. For example:
    • Coefficient of variation
    • Quantile
    • Median
    • Skewness coefficient
    • Kurtosis coefficient
  4. Convergence by probability & Convergence by distribution and theories behind these, like Law of Large Numbers.

Then the overall structure of mathematical statistic can be described as following:

  1. Basic definitions of mathematical statistic, include overall samples, individual sample.
  2. The three main Sampling distributions:
    1. X^2 distribution;
    2. F distribution;
    3. T distribution;
  3. Methods about Parameter Estimation, Interval estimation, Hypothetical test. This is the toughest and most important contents in this subject.

Here are two notes, and one is the overall review of this course which lists the main contents and points. Another is the practice of the questions in the textbook. I believe if you want to master this subject better, it would be necessary and an effective way to do more practices. Actually, both these two notes are somewhat simple, so, please reference the textbook and find more information. These notes would be useful during your review period.

So, here they are.

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