A little trap of the basic knowledge of mathematical statistics

A little trap of the basic knowledge of mathematical statistics First, Mathematical Expectations

Mathematical expectation is also called mean value, expectation, which is called expectation value in physics. In probability theory and statistics, the expectation of a discrete random variable is the probability of each possible result in the experiment multiplied by the sum of its results.

Defined:

The sum of all possible values of discrete random variables and their corresponding probability p is called mathematical expectation.

It is important to note that expectations are not necessarily equal to "expectations" in common sense-expectations may not be equal to every result. In other words, the expectation is the average of the output value of the variable, so the expected value is not necessarily included in the variable's output values collection.

Second, Variance ( Variance )

Variance is the average of the squares of the difference between each data and the average value. In probability and mathematical statistics, variance is used to measure the degree of deviation between a random variable and its mathematical expectation (mean value).

The average sum of squares of the difference between each data in the sample and the sample mean is called the sample variance, and the square root of the sample variance is called the sample standard deviation. Sample variance and sample standard deviation are measured by the amount of a sample fluctuation, the larger the sample variance or sample standard deviation, the larger the fluctuation of sample data, the more unstable.

Third, Standard deviation ( Standard Deviation )

The standard deviation, also known as the Mean variance (Mean square Error), is the average of the distance between the numbers deviating from the average, which is the square root of the mean squared and the average. The standard deviation is the square root of the variance of the arithmetic. The standard deviation can reflect the degree of deviation of a data set.

The standard deviation is also known as standard deviations, or the experimental standard is poor, the formula:

If it is overall, the standard deviation formula is divided by n within the square root, and if it is a sample, the standard deviation formula is divided within the square root (n-1), because we are exposed to a large number of samples, so the universal use of the square root is divided (n-1).

In simple terms, the standard deviation is a measure of the degree of dispersion of a set of data averages. A larger standard deviation represents a large difference between most values and averages; a smaller standard deviation that represents these values closer to the average.

Although the standard deviation has a unit of measure, and the variance has no unit of measure, the two act the same. The standard deviation uses the square method to eliminate the sign, so it is the most common and most important discrete trend statistic. The larger the standard deviation, the greater the difference between the values of the variables, the farther the data is from the average, the less representative the mean. Conversely, the smaller the standard deviation, the smaller the difference between the variables, the higher the average data distance, the more representative the mean.

Standard deviation applications and investments can be used as indicators to measure return stability. The larger the standard deviation value, the more volatile the return is from the past average value. Conversely, the smaller the standard deviation, the more stable the return, and the lower the risk.

Sample Standard deviation:

In the real world, it is unrealistic to find an overall true standard deviation unless in some special cases. In most cases, the overall standard deviation is estimated by randomly extracting a certain amount of sample and calculating the sample standard deviation.

From a large set of numerical X1, X2, ..., xn the same value combination x1, x2, ..., xn,n<n, often defines its sample standard deviation:

The sample variance is an unbiased estimate of the population variance. The denominator in S is n-1 (compared to the denominator in the population is N), because the degrees of freedom are n-1, because of the existence of constraints.

A little trap of the basic knowledge of mathematical statistics