勤勉233-优快云博客

原创【学习笔记】数理统计习题十二

Q1: Consider the multiple linear regression modelY=Xβ+ϵ,\boldsymbol{Y} = \boldsymbol{X}\boldsymbol {\beta} + \boldsymbol\epsilon,Y=Xβ+ϵ,where Y=(y1,…,yn)⊤\boldsymbol Y=(y_1,\dots,y_n)^\topY=(y1,…,yn)⊤, β=(β0,…,βp−1)⊤\boldsymbol\beta=(\beta_0,\dots,\b

2020-12-25 18:48:35 2051 1

原创【学习笔记】数理统计习题十一

Q1: Consider the linear modelyi=β0+β1xi+ϵi, ϵi∼iidN(0,σ2),i=1,…,n.y_i=\beta_0+\beta_1x_i+\epsilon_i,\ \epsilon_i\stackrel{iid}{\sim} N(0,\sigma^2), i=1,\dots,n.yi=β0+β1xi+ϵi, ϵi∼iidN(0,σ2),i=1,…,n.Derive the maximum likelihood estimator

2020-12-24 20:30:42 3054 1

原创【学习笔记】数理统计习题十

Q1: True or false, and state why:If the p-value is 0.03, the corresponding test will reject at the significance level 0.02.解：错误，如果α<p−value\alpha<p-valueα<p−value，则接受原假设If a test rejects at significance level 0.06, then the p-value is less

2020-12-24 18:46:56 2030 1

原创【学习笔记】数理统计习题九

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be an iid sample of N(μ,σ2)N(\mu,\sigma^2)N(μ,σ2), where μ\muμ is known. Show that this model has a monotone likelihood ratio. Given a significance level α\alphaα, derive a UMP test of the following hypotheses:H0:σ2≥σ

2020-12-22 21:45:46 2332

原创【学习笔记】数理统计习题八

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be an iid sample of Possion distribution with parameter λ>0\lambda>0λ>0. Find an approximate 100(1−α)%100(1-\alpha)\%100(1−α)% confidence interval for λ\lambdaλ.解：已知泊松分布的均值和方差均为λ\lambdaλ，X1,X2,⋯ ,XnX_1,X_2,\

2020-12-21 20:55:44 2368

原创【学习笔记】数理统计习题七

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be i.i.d. sample of X∼N(μ1,σ2)X\sim N(\mu_1,\sigma^2)X∼N(μ1,σ2), and Y1,…,YmY_1,\dots,Y_mY1,…,Ym be i.i.d. sample of Y∼N(μ2,σ2)Y\sim N(\mu_2,\sigma^2)Y∼N(μ2,σ2), where XiX_iXis and YjY_jYjs are independent, μ1,μ2

2020-12-20 17:04:12 2224 1

原创【学习笔记】数理统计习题六

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be an iid sample of Poisson distribution with parameter λ\lambdaλ. Which of the following are unbiased estimates of λ\lambdaλ? ( ABCABCABC ) 多选A. Xˉ\bar XXˉB. Sn∗2S_n^{*2}Sn∗2C. (Xˉ+Sn∗2)/2(\bar X+S_n^{*2})/2(Xˉ+

2020-12-18 20:15:05 3759 1

原创【学习笔记】数理统计习题五

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be a sample from densityf(x;σ)=12σe−∣x∣/σ,f(x;\sigma)=\frac{1}{2\sigma} e^{-|x|/\sigma},f(x;σ)=2σ1e−∣x∣/σ,where σ>0\sigma>0σ>0. Find the maximum likelihood estimator for σ\sigmaσ.Solution：似然函数为：L(σ)=∏i=1n

2020-12-10 16:24:00 1933

原创【学习笔记】数理统计习题四

Q1: Let X1,…,XnX_1,\dots,X_nX1,…,Xn be an iid sample of N(1,22)N(1,2^2)N(1,22). Which of the following items are true? ( D )A. Xˉ−12/n∼t(n)\frac{\bar X-1}{2/\sqrt{n}}\sim t(n)2/nXˉ−1∼t(n)B. 14∑i=1n(Xi−1)2∼F(n,1)\frac{1}{4}\sum_{i=1}^n(X_i-1)^2\sim F

2020-12-09 23:04:10 2159 1

原创【学习笔记】数理统计习题三

Q1: Let X1,…,X15X_1,\dots,X_{15}X1,…,X15 be a simple random sample of N(0,22)N(0,2^2)N(0,22). What is the distribution ofY=X12+⋯+X1022(X112+⋯+X152)?Y=\frac{X_1^2+\dots+X_{10}^2}{2(X_{11}^2+\dots+X_{15}^2)}?Y=2(X112+⋯+X152)X12+⋯+X102?Solution：因为Xi

2020-12-09 13:06:28 2090

原创【学习笔记】数理统计习题二

Q1: Assume that X1∼Gamma(α1,λ)X_1\sim Gamma(\alpha_1,\lambda)X1∼Gamma(α1,λ), X2∼Gamma(α2,λ)X_2\sim Gamma(\alpha_2,\lambda)X2∼Gamma(α2,λ), and they are independent. Prove that:Y1=X1+X2∼Gamma(α1+α2,λ)Y_1=X_1+X_2\sim Gamma(\alpha_1+\alpha_2,\lambda)Y1

2020-12-08 16:38:25 2727

原创【学习笔记】数理统计习题一

Q1: The planet Tralfamadore has years with 500 days. There are 5 Tralfamadorans in the room. Write an expression for the probability that no two of them have the same birthday.Solution: Suppose that there are n Tralfamadorans in the room. The probability

2020-12-07 14:31:23 2678 1

原创【学习笔记】Miller素数测试

根据费尔马定理，如果n为素数，则对所有不被n整除的a，都有an−1≡1(modn)a^{n-1}\equiv1(mod n)an−1≡1(modn)成立，如果不成立，说明是合数，但是，如果成立，不能排除n是合数的可能性。因此还要进一步进行检测。如果n是素数，则n-1必然是偶数。因此，可令n−1=2qmn-1=2^qmn−1=2qm，并考察下面的测试序列：am(modn)a^m(mod n)am(modn) , a2m(modn)a^{2m}(mod n)a2m(modn) , a4m(modn)a^{4

2020-06-09 22:02:42 486

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