Expectation: In probability theory and statistics, the expectation of a discrete random variable is the sum of the probability of each possible result in the experiment multiplied by the value of this result.

If the probability of each experimental result is assumed to be equal, then expectation is the average of equal probability "expectation" calculated by adding the results of repeated random experiments under the same opportunity. It should be noted that the expected value may not be equal to every result, because the expected value is the average of the output values of variables, and the expected value is not necessarily included in the set of output values of variables.

The expected formula of discrete random variables is as follows, assuming that the random variable is XX, take the value Xi (I = 1, 2 ..., n) Xi (I = 1, 2, ..., n), corresponding to the occurrence probability Pi (I = 1, 2, ..., n).

When pi (I = 1, 2, ..., n) pi (I = 1, 2, ..., n) is equal, that is, when pi= 1npi= 1n, E(X)E(X

The expectation of a continuous random variable can be obtained by integrating the value of the random variable with the corresponding probability product. Let XX be a continuous random variable and f(x)f(x) be the corresponding probability density function, then E(X)E(X) is expected to be: E (x) = ∫ XF (x) DXE (x) = ∫.

Variance: In probability theory and mathematical statistics, variance (symbol D, or σ2σ2) is used to measure the deviation between a random variable and its mathematical expectation (i.e. mean value). In calculation, variance is the average of the sum of squares of the difference between each data and its average.

Variance is a standard to measure the degree of data dispersion, which is used to indicate the degree of deviation between data and data center (mean). The greater the variance, the greater the degree of data deviation from the center. At the same time, the expectations of variables are the same, but the variance is not necessarily the same.

Still taking the discrete random variable as an example, assuming that the random variable is XX, taking the values Xi (I = 1, 2, ..., n) Xi (I = 1, 2, ..., n) and μ μ is the mathematical expectation (mean) of the random variable, then the variance of the discrete random variable XX can be expressed as: d (. μ)2D(X)= 1n∑i= 1n(xi？ μ)2。

In the calculation, if the expected E(X)E(X) of the random variable XX is known, the calculation of variance can be simplified as: D(X)=E(X? E(X))2=E(x2)？ [E(x)]2D(X)=E(X？ E(X))2=E(x2)？ [English (x)]2.