Online notes on Mathematics for postgraduate entrance examination (9): a series of courses on probability theory and mathematical statistics

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Because it can't be typed , So in this article, we use A^- Instead of the inverse of an event

1 Random events and probabilities

Think of events as collections , Don't take it as an event

Formula list

Not independent ,P(A+B) And P(A∪B) Exactly the same

P(A-B) And P(AB^-) And P(A)-P(AB) Exactly the same ; In the case of Independence ,P(AB) And P(A∪B) Exactly the same

Common counterexamples ： The complete / An empty set 、 The two events are equal 、 The two events are mutually exclusive

Sufficient and necessary conditions , You need four certificates , If you can't prove it, it's in A and B Mongolia

Additive multiplication （ Merge ） There is a rate of distribution from time to time P(A∩(B∪C))=P((A∩B)∪(A∩C)), But time reduction does not P(A∩(B-C))≠P((A∩B)-(A∩C))

A^-∩B^-=(A∪B)^-; The operation of an event cannot be regarded as a numerical operation

2 One dimensional random variables and distributions

The distribution function is required to be nonnegative , And F(+∞)=1,F(-∞)=0; The distribution function requires right continuity <–> When the limits on both sides of the dividing point are not equal , The interval should be written as left closed and right open ; The distribution function is required to be monotonic

The standard normal is about y axial symmetry

The derivative of the distribution function is the density function ;F(X)={X≤x},X It can be replaced by such as -X And other random variables

Piecewise distribution function , Are reduced to less than or equal to or less than the sign , The less than or equal sign is used F(X), The less than sign uses the left limit

3 Two dimensional random variables and their distribution

Add and subtract the independent normal distributions , Expect to add and subtract , Variance addition ; Expectation is the axis of symmetry of the normal distribution function

The correlation coefficient of two-dimensional normal distribution is 0, The two one-dimensional normal distributions on its edge are independent ; Two normal distributions 、 Other distribution correlation coefficients are 0, Not necessarily independent

Independence , The distribution function of the joint distribution is the product ; similar X<Y In combination with the image, the second integral is used to calculate

The marginal distribution is the respective distribution ; Each is identically distributed , Together, they are not necessarily identically distributed

4 Digital features

Expected variance of common distribution function

DX=EX²-(EX)²

When encountering the distribution function to find the numerical characteristics , Find the derived density function ;E(g(x))= Density function and g(x) Integral ; Odd functions in (-∞,+∞) The anomalous integral of is 0

Parameter is λ The expectation of Poisson distribution of is λ

Covariance is linear as expected （ Variance only exists when it is independent ）; Two random variables are independent , The covariance is 0; Take covariance with yourself , That's variance

The two variables are uncorrelated , The covariance is 0,EXY-EXEY=0; Expected to be linear

If there is a plus sign in the covariance , You can perform their own cross calculation

iid Is independent and identically distributed

The correlation coefficient is 0 ±1 When there is a and b bring aX+B The probability of is 1（1 when a>0,-1 when a<0）; The first of the normal distributions is the expectation , The second is the square of the variance

Discrete joint distribution , Can be calculated by column distribution column

It's a question to investigate the application of formulas

5 The law of large numbers and the central limit theorem

The essence of Chebyshev inequality is variance , Only one formula is used in the final answer

Sinchin's law of large numbers （ It is required to be independent and identically distributed + Have expectations ）

The problem of central limit theorem is standardization in essence ： Subtract expectations , Divided by standard deviation （ Variance open root sign ）

6 mathematical statistics

Sample variance is an unbiased estimate of population variance ; The mean of the sample variance is the population variance

The chi square distribution is n（ freedom ） Sum of squares of standard normal distributions ;t The distribution is a standard normal distribution divided by n Is the chi square distribution of degrees of freedom divided by n Square root ;F Distribution is the quotient of two chi square distributions divided by degrees of freedom

All three distributions are required to obey independent identically distributed ; The square of a single normal also obeys chi square

The degree of freedom is n Chi square distribution of , The expectation is n, The variance of 2n; The degree of freedom is n The chi square distribution of follows 1/n The exponential distribution of （ Cold knowledge ）

There are two distributions at the same time , Find the relationship between them by definition ;T ~ t(n), be T² ~ F(1,n);t The density function of the distribution is about y axial symmetry

The sampling distribution of a normal population

The mean value of normal distribution directly corresponds to addition and subtraction , Variance directly corresponds to addition ; Conclusion on the variance of single distribution and mean distribution and associated samples

A normal population has its own independent condition , Variance can be directly calculated by linear method ; In the case of normal population, find S² The variance of , The degree of freedom of forced construction is (n-1) Chi square distribution of

Such as the variance of the square of the mean value , Can be forced into the standard type , Construct chi square distribution

C The formula of the option is (n-1)S²; The variance of the mean is σ²/n

When the first moment is not available （ There was no θ） Consider using the second moment

The maximum likelihood estimate of the discrete function is multiplied by the corresponding probability of the observed value , Take the logarithmic derivative order as 0 seek θ that will do

Maximum likelihood estimation of discrete functions The amount You need to set the number of values