mathematical statistics with resampling and r

Introduction"A more official introduction" Mathematical statistics is a subject based on probability theory and highly applied. It studies how to collect, collate and analyze the data with randomness in an effective way so as to make the correct inference and prediction for the problems examined, and provide the basis and suggestion for the correct decision-making and action.

A little trap of the basic knowledge of mathematical statisticsFirst,Mathematical ExpectationsMathematical expectation is also called mean value, expectation, which is called expectation value in physics. In probability theory and statistics, the expectation of a discrete random variable is the probability of each possible result in the experiment multiplied by t

In probability theory, the stochastic variables are assumed to be known, and the properties and digital characteristics of the study are studied.In mathematical statistics, the distributions of the random variables studied are unknown or not fully known, and many observations are obtained by repeating independent experiments to infer the various possible distributions of random variables.1. Random samplesOv

Chapter I. Stochastic events and probabilitiesChapter two stochastic variables and their distributionsChapter three multivariate random variables and their distributionsThe fourth chapter law of large numbers and the central limit theoremThe fifth chapter statistic and its distributionThe sixth chapter parameter estimationThe seventh chapter hypothesis testEighth chapter analysis of variance and regression analysisChapter I. Stochastic events and probabilities1.1 Random events and their operatio

{\BF Question 2:\}Suppose that $X $ have density function\[F_\theta (x) = \frac{x^{\theta-1} e^{-x}}{\gamma (\theta)} I \{x>0\}.\]Find expressions for the mean and variance of $\log x$.{\BF Solution:\}Rewriting, we obtain\[F_\theta (x) = \exp\{\theta \log x-a (\theta) \} h (x)\]For $A (\theta) = \log\gamma (\theta) $ and$h (x) = x^{-1}e^{-x}i\{x>0\}$. This is a canonicalExponential family with sufficient statistic $\log x$,Which means that\[E \log X = \frac{d}{d\theta} A (\theta) =\frac{\gamma '

, then X1 and X2 only one of them, the greater the correlation between features the greater the absolute value of the correlation coefficient, the correlation coefficient matrix can be used to filter the characteristics.Estimating parameters with samples Moment estimation Moment estimation method, also known as "moment method Estimation", is to use the sample moment to estimate the corresponding parameters in the whole. The simplest method of moment estimation is to estimate the to

? Sigma Σσς Tau Ττ Upsilon Υυ Phi Φ?φ Chi Χχ Psi Ψψ Omega Ωω Sampi ?? Reference:1. Greek alphabet for Mathematics, science and engineering: HTTP://ZH.WIKIPEDIA.ORG/WIKI/%E7%94%A8%E6%96%BC%E6%95%B8%E5%AD%B8%E3%80%81%E7%A7%91%E5%AD%B8 %e5%92%8c%e5%b7%a5%e7%a8%8b%e7%9a%84%e5%b8%8c%e8%87%98%e5%ad%97%e6%af%8d2.

variables, the x and Y correlations are equivalent to X and Y Independent.Seven, covariance matrix:Set n random variables (x1,x2,.... Xn), Cij=cov (XI,XJ) is present, then it is called matrix is the covariance matrix. Because of cij=cji, the above matrix is the match matrix.Viii. Upper bound of covarianceWhen and only if x, Y have a linear relationship, the equals sign is established.Nine, correlation coefficient:Ten, MomentFor the random variable x,x, the K-Order Origin moment is:The K-Order c

:5SOURCE Analysis: Direct violence can code:#include using namespaceStd;typedefLong LongLL;#defineMAX_V 105intMain () {intT; intn,m; scanf ("%d",t); while(t--) { intk=1; while(~SCANF ("%d%d",n,m)) {if(n==0m==0) Break; intC=0; //Direct violence can for(intI=1; i1; i++) { for(intj=i+1; j) { if((i*i+j*j+m)% (i*j) = =0) C++; }} printf ("Case %d:%d\n", k++, c); } if(t!=0) printf ("\ n"); } r

, Bayesian estimation11, Moment estimation: The estimation of the total corresponding parameters by the estimation of the sample moment (note the estimate of variance using the second-order center moment of the corrected sample) "parameter is not necessarily expected and variance, such as evenly distributed parameters"PS: Can be used low-order moment is not high-order, such as the Poisson distribution of the mean and variance is the same parameter, at this time should use the first order of the

constant solution in the plural range, the number is the number of the equation (the root is calculated by the weight number), so the N-order matrix A has n eigenvalues. 3.2 Solving eigenvalues and eigenvectors Examples: 3.3 properties 1) The characteristic vectors belonging to different eigenvalues are linearly independent. 2)λ3)similarity: is the equivalent matrix before Set A, B are n-order matrices, if there is a full rank matrix p, so that