catalogue
Overview of discrete type
successive type
Definition of mathematical expectation 1:
Definition 2:
Calculate the mathematical expectation value of random variables
Mathematical expectation algorithm for individual data
Overview of discrete type
successive type
Definition of mathematical expectation 1:
Definition 2:
Calculate the mathematical expectation value of random variables
Mathematical expectation algorithm for individual data
Expand and edit this paragraph overview.
Mathematical expectation The mathematical expectation of discrete random variables.
Discrete type
The sum of the products of all possible values xi of a discrete random variable and the corresponding probability Pi(=xi) is called the mathematical expectation of the discrete random variable (let the series converge absolutely), and it is recorded as E(x). One of the most basic mathematical characteristics of random variables. It reflects the average value of random variables. Also called expected value or average value. If a random variable only gets a limited number of values, it is called the mathematical expectation of discrete random variables. It is a generalization of simple arithmetic average, similar to weighted average. For example, a city has 65,438+10,000 families, 65,438+10,000 families have no children, 90,000 families have one child, 6,000 families have two children, and 3,000 families have three children. Then the number of children in any family in this city is a random variable, which can be taken as 0, 1. The probability of taking 1 is 0.9, that of taking 2 is 0.06, and that of taking 3 is 0.03. Its mathematical expectation is 0× 0.01+/kloc-0 /× 0.9+2× 0.06+3× 0.03, which is equal to 1. 1658.
successive type
The probability density function of continuous random variable x is f(x), if the integral:
Absolute convergence, then this integer value is called the mathematical expectation of random variable x, and it is recorded as:
Edit the definition of mathematical expectation in this paragraph.
Definition 1:
Expected value of mathematics
According to the definition, the sum of the products of all possible values of a discrete random variable and its corresponding probability p is called mathematical expectation, which is denoted as E. If a random variable only obtains a limited number of values: x, y, z, ..., it is called a discrete random variable.
Definition 2:
1 The conventional safety factor is selected according to experience, that is, the ratio of the average limit strength of materials (called mathematical expectation in probability theory) to the average working stress (mathematical expectation).
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Mathematical expectation value of random variables
Mathematical expectation in probability theory
And statistics, the expected value (or mathematical expectation, or mean, or expectation for short) of a discrete random variable is the sum of the probability of every possible result multiplied by its result in the experiment. In other words, the expected value is the average of the equivalent "expectations" calculated by repeating the results of random experiments under the same opportunity. It should be noted that the expected value is not necessarily equal to the common sense "expectation"-"expected value" is not necessarily equal to every result. (In other words, the expected value is the average of the output values of variables. The expected value is not necessarily contained in the set of output values of variables. )
Mathematical expectation algorithm for individual data
The definition of mathematical expectation is this. Mathematical expectation e (x) = x1* p (x 1)+x2 * p (x2)+...+xn * p (xn) x1,x2, x3, ... where xn is these data and p (x/klk). The probability function of P (X 1), P (X2), P (X3), ... P (Xn) is understood as the frequency of data X 1, X2, X3, ... Xn is f(Xi), then e (x) = x1.
It is easy to prove that E(X) is the arithmetic mean of these data. Let's give an example. For example, several numbers: 1, 1, 2, 5, 2, 6, 5, 8, 9, 4, 8, 1 appear three times, accounting for 3/ 12 of all data. Similarly, we can calculate F (2) = 2/ 12, F (5) = 2/ 12, F (6) =1/2, and F (8) = 2//kloc-0. F (4) =112 according to the definition of mathematical expectation: e (x) =1* f (1)+2 * f (2)+5 * f (5)+6 * f (6)+. Now calculate the arithmetic mean of these numbers: xa = (1+1+2+5+2+6+5+8+9+4+8+1)12 =13.