1.2. 1 random events and sample space
The set of all possible results of uncertain events constitutes the sample space of random events, and each concrete result in the sample space is called a random event in the sample space. To deeply understand the concept of probability, we must first know the related properties of frequency. Generally speaking, if the number of times a random event appears in n tests or observations is nA, it is called Na.
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Is the frequency of event a in these n tests or observations. Although the frequency of event A is a variable in many observations, the observation of many physical phenomena shows that fn(A) swings around a constant when the number of experiments or observations n increases gradually, and gradually stabilizes at this constant, that is to say, the frequency is stable. The stability of frequency is of great significance for us to understand the inherent regularity of random phenomena, predict things and control things.
For random event A in sample space S, the frequency in n trials has the following properties.
( 1)0≤fn(A)≤ 1
(2)fn(S)= 1
Based on the understanding of the concept of frequency, suppose that E is a random test, S is all sample spaces of the test, and give a real number P(A) to each specific event A in the test, then P(A) is called the probability of occurrence of event A, if the following conditions are met:
( 1)0≤P(A)≤ 1
(2)P(S)= 1
(3) For the paired incompatible event AK (k = 1, 2, ...), there are also:
P(a 1∪A2∪…∪An∪…)= P(a 1)+P(A2)+P(A3)+…+P(An)+…
Then this probability is said to be countable and additive. See reference [53] for the algorithm of probability.
1.2.2 random variable
In order to comprehensively study random events and analyze the inherent regularity of random problems and reveal the statistical regularity of uncertainty or random problems in the objective world, it is necessary to understand the basic concepts of random variables.
Let e be a random experiment, and its sample space is S={e}. If there is a real number X(e) in the sample space corresponding to a specific random event e∈S, then every E in the space S always has a real-valued single-valued function X(e), that is, the generated S corresponds to the function of X(e), and X(e) is called a random variable.
Let x be the whole of all possible values of X(e), then there is the following schematic relationship (Figure 1.7):
Because random variables are functions of random events, the occurrence of random events has a certain probability. Therefore, the value of random variables also has a certain probability, which shows that there is an essential difference between random variables and ordinary functions. Ordinary functions are defined on the real number axis and random variables are defined on the sample space (the sample space elements are not necessarily real numbers).
Figure 1.7
It is common to define a real function on the sample space S={e} to form a random variable. As shown in table 1. 1, the hydrogeological parameters are a group of random variables, and an observation of hydrogeological data (a random event) is realized, and a group of hydrogeological parameters (random variables) can be obtained according to a certain functional relationship. The introduction of random variables is mainly to help us analyze and study random problems by mathematical analysis.
Random variables can be divided into discrete random variables and continuous random variables. The so-called discrete random variable means that all possible values are finite or infinitely countable.
Generally speaking, let all possible values of the discrete random variable X be xk (k = 1, 2, …), and the probability of taking each possible value of X is:
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Pk shall meet the following two conditions:
( 1)Pk≥0 k= 1，2，…
(2)
The formula P{X=xk}=Pk is called the probability distribution or distribution law of discrete random variables, and the probability distribution of common discrete random variables is as follows.
(1)(0- 1) distribution. A random event has only two possible outcomes, that is, its sample space only contains two elements S={e 1, e2}, so we define a random variable.
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To describe and characterize this kind of random problem and call it (0- 1) distribution.
(2) binomial distribution. Let a random event have only two possible outcomes, S={e 1, e2}. If the probability of event e 1 is p, and the probability of event e2 is 1-p, then there is p {x = e1} = p.
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If the Bernoulli test is carried out for n times for the above random problems, the event e 1 may occur 0, 1, 2, …, n times. Through calculation, it is not difficult to find that the probability that the event e 1 happens exactly k times (0≤k≤n) is:
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Note that it happens to be the k+ 1 term in the binomial (p+q)n expansion, so we say that the random variable x obeys the binomial distribution of parameters n and p, and is denoted as x ~ b (n, p).
(3) Poisson distribution. Let all possible values of the random variable X be 0, 1, 2, … The probability of taking the kth value is, k = 0, 1, 2, … where λ > 0 is a constant, then X is said to obey the Poisson distribution with parameter λ. Let it be x ~ π (λ). ( 1.6)
Continuous random variable and its probability density: There is a random variable X whose distribution function is F(X). If there is a nonnegative function f(x), for any real number, there are:
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Then x is called a continuous random variable, and f(x) is called the probability density function of x, which can be abbreviated as probability density. F(X) is called the distribution function of x, and the distribution function of continuous random variables is also a continuous function.
The probability density function reflects the relative probability of a specific random event in the sample space, while the distribution function of random variables reflects the probability of random events in a specific area or time domain. The probability density function f(x) has the following basic properties.
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Graphs 1.8 to 1. 1 1 reflect the basic meanings of probability density function and probability distribution function of random variables.
Several common distributions of important continuous random variables are as follows.
(1) evenly distributed. If the continuous random variable X takes a value in a certain interval (a, b), its probability density function is:
Figure 1.8
Figure 1.9
Figure 1. 10
Figure 1. 1 1
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It is said that x obeys the uniform distribution on (a, b), and its distribution function is:
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(2) Normal distribution. If the probability density of continuous random variable x is:
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Where: μ, σ-constant. X- obeys the normal distribution with parameters μ and σ. Typical schematic diagrams of density function and distribution function of normal distribution random variables are shown in figure 1. 12 and figure 1. 13.
Figure 1. 12
Figure 1. 13
From the formula (1. 10) and the figure 1. 12, it can be seen that μ and σ are important parameters to describe normal distribution random variables, μ reflects the maximum probability position of random variables on (-∞,+∞), while σ reflects (-∞,+∞).
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1.2.3 Numerical characteristics of random variables
Although the probability density function or distribution function of a random variable can well describe and depict the basic characteristics of random variables, it is often difficult to know the specific distribution function formula of random variables encountered in production practice. However, through the statistical analysis of random variables, some important numerical characteristics reflecting the properties of random variables, such as mathematical expectation, variance, moment and so on. , will get.
If the distribution law of discrete random variable x is:
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And it is absolutely convergent, then E(X)= is the mathematical expectation of a random variable.
If x is a continuous random variable, its probability density function is f(x) and is integrated.
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As can be seen from the definition of mathematical expectation of random variables above, its physical meaning is equivalent to weighted average. The definition of the mathematical expectation of a function of a random variable is similar to that of a random variable, and the mathematical expectation of a random variable has the following important properties:
(1) Let c be a constant, then e (c) = c
(2) Let X be a random variable and C be a constant, then e (CX) = c e (x).
(3) Let x and y be any two random variables, then E(X+Y)=E(X)+E(Y).
(4) If x and y are two independent random variables, then E (X Y) = E (X) E (Y).
The average value of random variables only reflects the average level of random variables, and it is difficult to describe the deviation degree of each individual from the average level of random variables. In order to study and analyze the deviation of random variables from the mean, it is necessary to introduce the concept of variance of random variables.
Let x be a random variable, and if E{[X-E(X)]2} exists, then E{[X-E(X)]2} is the variance of x, denoted as D(X) or var(X), that is:
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It is not difficult to see from the above formula that variance is actually the concept of square difference. If the other party divides by the square root, you can get the mean square deviation or standard deviation, which is recorded as σ(X):
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The following is an important formula for calculating the variance of random variables:
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The variance of random variables has the following important properties:
(1) Let c be a constant, then D(C)=0.
(2) let x be a random variable and c be a constant, then D(CX)=C2D(X)
(3) Let x and y be two independent random variables, then D(X+Y)=D(X)+D(Y).
See table 1.2 for the numerical characteristics of several common distributed random variables.
1.2.4 Covariance and correlation coefficient
The previous section introduced the numerical characteristics of a random variable, but in practical engineering, there are often two or more random variables, and different random variables have some correlation. In order to study the relationship between different random variables, it is necessary to understand the concepts of covariance and correlation coefficient.
Let x and y be random variables, then the covariance between x and y is:
but
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Table 1.2
Is the correlation coefficient or standard covariance of x and y, and the covariance is calculated as follows.
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Covariance has the following properties:
( 1)cov(X，Y)=cov(Y，X)
(2)cov(aX，bY)=ab cov(X，Y)
(3) cover (X 1, X2, Y)= cover (X 1, Y)+ cover (X2, y)
Similarly, for the random variable X, Y, if e (xk), k= 1, 2, … exists, it is called the k-order origin moment of X. If e [(x-e (xk))], k= 1, 2, …] exists, it is called the k-order of X. It is called the k+l-order central mixing moment of X and Y. It is not difficult to see that the mathematical expectation E(X) of random variable X is the first-order origin moment of X, while its variance is the second-order central moment and covariance is the second-order central mixing moment of random variable.
The related properties of two-dimensional random variables can be directly extended to n-dimensional random variables, among which the covariance matrix of n-dimensional random variables is the most commonly used:
Let (X 1, X2, …, Xn) be an n-dimensional random variable, and the second-order central moment between the two variables is:
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It is called a matrix:
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Is the covariance matrix of n-dimensional random variables. According to the property of covariance of random variables Cji=Cij, matrix C is a symmetric matrix.