Stochastic and Normally Distributed Probability Distributions Allow for Statistical Analysis and Modelling of Returns

Problem 1 Probability distribution of highly risky asset returns is assumed to be random with revenue, mainly on average revenue. This assumption enables statistical analysis and reward modeling. The model assumption is as follows. i) There is no transaction cost to purchase or sell assets. As trading costs are usually the smallest percentage of investments, they are less important to investors and reduce the complexity of the capital asset pricing model (CAPM).

In stochastic theory, the normal (or Gaussian or Gaussian or Laplacian-Gaussian) distribution is a very common continuous probability distribution. Normal distribution is important in statistics and is often used in natural science and social science to represent real-valued random variables whose distribution is unknown. Random variables with Gaussian distribution are called normal distributions and are called normal deviations. Normal distribution is more useful than the central limit theorem. In the most general form, under a certain condition (including finite variance), when the average value of observation samples of random variables independently derived from independent distributions converges to a normal distribution, that is, when a number is observed It becomes positive. The state distribution is large enough. Physical quantities (eg measurement errors) that are expected to be the sum of many independent processes are usually nearly normal.

Measurement errors in physical experiments are usually modeled by normal distribution. The use of this normal distribution does not mean to assume that the measurement error is a normal distribution but rather uses the normal distribution to generate the most reliable prediction and only gives knowledge about the mean and variance of the error I will. In a standardized test, you can select the number and difficulty of the problem (as in the case of the IQ test), or by fitting the score of the original test to the normal distribution to fit the score of the original test It can be normalized. . For example, the conventional 200 to 800 range of SAT is based on a normal distribution with average 500 and standard deviation 100.

In probability theory and statistics, the chi-square distribution with k degrees of freedom (chi-square or χ 2 distribution) is also the distribution of the sum of squares of k independent standard normal random variables. Chi square distribution is a special case of gamma distribution, and is one of the most widely used probability distributions in inference statistics, especially in hypothesis testing and construction of confidence intervals. When it is distinguished from the more general non-centered chi-square distribution, this distribution is sometimes called the central chi-square distribution.