I. Classical and geometrical approximate 1.1 classical and geometrical approximate features 1) in common: equal probabilities (the probability of each occurrence being the same) 2) difference:
- The classical approximate sample space is a finite set.
- A geometric profile can be an infinite set, but it can be represented by a geometric region
1.2 equation 1) Classical Overview: The number of basic events known as N and the result of the event A is a few m, and then substituting the formula:
That is, the probability of event A.
2) Geometrical Overview: The use of geometric regions with measures (length, area, volume, etc.) is indicated:
1.3 Solution Steps 1.3.1 Classical Approximate type1) Judge whether the event is an equal probability event, and use the letter to denote the event;
2) Using enumeration method to calculate the number of basic events N and the number of basic events contained in event a m;3) calculates the probability of a in the event of a 1.3.2 geometric approximate 1) the sample space and the probability of the event is expressed by the relationship, which is divided into two categories: A, the sample space has the obvious geometrical meaning, the sample point is located in the Geometry Region topic has given B, the geometric region of the event of the sample space is not given directly, to find out that they become the key to solve such geometric probability problems. The method is to introduce the variables first, then use algebraic formula to express the relationship between the variables, and then draw the solution according to the geometrical shape. 2) Draw the geometry in the coordinate system 3) according to the image according to the classical approximate formula solution Example:
Second, conditional probability and Bayesian 2.1 define 2.1.1 Conditional probability
2.1.2 Multiplication Formula
This leads to:
Cases:
Note: When a belongs to B, P (a) =p (AB)
2.1.3 Full probability formula: 1) Principle: The multiplication formula extension is set to two events, then it can be expressed as
Obviously, if the
This can be done by:
2) Definition:
Set the test E The sample space , For the event, for a split, and , the
The above equation is called the full probability formula .
Cases:
For
2.1.4 Bayesian formula
set the sample space for the experiment, for the event, for a split, and , the
The upper type is called the Bayesian formula.
Proof:
2.1.5 Independent Event 1) two events IndependentNote: At this point P (a| B) =p (A) =p (AB)/P (B)2) independent of multiple events 3) N-heavy Bernoulli test (n-Re-independent repetition test) two-term theorem: For Bernoulli generalization, the probability that event a will occur k times in n trials
P (b|) in the probability of 2.2 pieces A) characteristics (full probability formula and Bayesian formula)Common denominator: All are generalized by conditional probability and multiplication formula.
difference:
- The full probability formula solves P (B), where a is decomposed into a=a1+a2+ ... The set of an, which decomposes P (B) =p (A1B) +p (A2B) + ... P (AnB) to solve.
- The Bayesian formula solves P (bi| A), which is called a posteriori probability, P (a) is called a priori probability. It can be understood that there are many factors that cause p (a) to occur, B=B1+B2+...+BN, in which the probability that the bi causes A to occur is the posteriori probability. It can be used to analyze the importance of various preconditions.
Three, the law of large numbers and the central limit theorem 3.1 large number theorem the arithmetic mean μ asymptotic 3.2 of multiple random variables the central limit theorem when n is sufficiently large, independent of the random variable with the distribution X1,x2,...,xn, and E (xi) =μ,d (xi) =σ2> 0, then these random variables and obey the normal distribution:
These random variables and the mean values are normally distributed as follows:
and four, parameter estimation and hypothesis test,
4.1 parameter estimate 4.1.1 point estimate 1) moment estimate 2) Maximum likelihood estimate 4.1.2 estimate evaluation criteria 1) unbiased 2) Validity 3) Consistency 4.1.2 Interval estimation confidence level (confidence): 1-α
4.2 Hypothesis test Significant level:α, take 0.05, 0.01 and 0.1
4.3 The similarities and differences of parameter estimation and hypothesis test 4.3.1 Common principle is obtained by the following normal distribution law:
4.3.2 Differences
- The parameter estimates assume that the mean value x (BA) falls into the Horizontal axis range (-Zα/2,zα/2 ) The probability is 1- α
This results in an estimated interval:
- hypothesis test that x (BA) falls into the horizontal axis range (0, - Z Α/2 ) and ( Z Α/2 , 0) events are small probability events, for a given small probability α (0< α <1) are:
If
deny Domain) was established, then rejected the original hypothesis H0, receive H1, otherwise there is no sufficient reason to refuse H0, should be recognized H0. 4.3.3 Solution Steps
1. Select a shaft armature: distribution known z (x1,x2,...,θ) 2, determine confidence interval: p{-ZΑ/2<Z (x1,x2,...,θ) <ZΑ/2}=1-α3. Simplification gets:p{θ1(X1,x2,...,xn) <θ<θ1(X1,x2,...,xn)}=1-α, thenget the interval estimation of the parameters (θ 1,θ 2)
1, proposed the original hypothesis H0, as well as the alternative (optional) hypothesis H1. (where H0 and H1 are opposites) 2, set the original hypothesis, and to construct a small probability of the event, the probability value of p=α3, the sample data to determine whether the small probability event occurred, if the occurrence of the refusal of H0, recognized H1.
Attached: Permutation combination formulaEquation Description: A (N,M) in the formula is the permutation number formula, C (N,M) is the combined number formula.
References:1, Liu Anping, Shohaijun, etc.,"Probability theory and Mathematical Statistics", Science press2. Lectures on probability and mathematical statistics of Zhengzhou Institute of Light Industry:Http://lxy.cumtb.edu.cn/gailvtongjidaoxue/gailvlunyushulitongjizhidao.htm
From for notes (Wiz)
Probability statistics of "mathematical Statistics"
