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

If the distribution function F(x) of a random variable X can be expressed as an integral of a non-negative integrable function f(x), then X is called a continuous random variable, and f(x) is called a probability density function (distribution density function) of X. Random variables whose range is one or several finite or infinite intervals can be listed one by one in a certain order. Discrete random variables and continuous random variables are also determined by the range (or in the form of values) of random variables. Variables can only take discrete natural numbers, that is, discrete random variables For example, if you toss 20 coins at a time, k is a random variable, and the value of k can only be the natural number 0, 1, 2, ..., 20, but not the decimal number 3. If a variable can take any real number in a certain interval, that is, the value of the variable can be continuous, then this random variable is called a continuous random variable.

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 value of a single 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.