北京理工大学数学专业试验设计期末试题(MTH17090)

课程编号：07000237 北京理工大学2011-2012学年第二学期2009级应用回归分析期末试题A 卷1.(35)Consider the following model:0112233i i i i i y x x x ββββε=++++,where y=labor force paticipation (%)by family heads of poor families, x 1=mean family income ($), x 2=mean family size,x 3=unemployment rate (% of civilian labor force unemployed).Two versions of the model were estimated as follows (the standard errors are in the brackets).(A)123ˆ33.460.01915.520.813i i i i yx x x =-+++ (48.78) (0.019) (9.46) (1.911)()Re 15,5130.13,3716.98T s n SS SS A ===(B) 12ˆ26.510.01815.30i i i yx x =-++ (44.37) (0.018) (9.12)()Re 3778.11s SS B =(1)Interpret the coefficient of mean family income in model (B);(2)Carry out a t-test to test whether in model (A) mean family size has a significant effect upon labor force paticipation;()0.05α=(3) Carry out a partial F-test to test whether in unemployment rate has a significant effect upon labor force paticipation;()0.05α=(4)What is the adjusted coefficient of determination 2R in model (A); (5)Test the significance of model(B);()0.05α=(6)Find a 95% confidence interval for the coefficient 1β of 1x in model (B); (7)Interpret the confidence coefficient 95% in (6).x 1=national income (100 million yuan) x 2=volume of consumption (100 million yuan) x 3=volume of passengers on railway (ten thousands persons) x 4=length of airline of civil aviation (ten thousands persons) x 5=number of inbound tourist arrivals (ten thousands persons) y=volume of passengers of civil aviation (ten thousands persons)(1)What problem do the VIFs imply? (2)Which regression coefficients may have the wrong sign? (3)Discuss the reasons for the problem in (2).3.(12)Consider the following model (n=8):2012y x x βββε=+++where y=body temperature of a pig (centi) x=time length after the pig is infected (hours)(1)Test the significance of 2x ;()0.05α= (2)Predict body temperature at x=80; (3)If the observations of x lie in (8,64),what ’s your suggestion about the prediction in (2); 4.(18)()()()2,:,,0,,0y X X n p rk X p E Var V V βεεεσ=+⨯===>, (1)Find GLSE for β;(2)Find an unbiased estimator for 2σ.5.(20)Full model ()()112220,,1,2,,,cov ,0,i i i i i i j y x x E i j n i ji j ββεεσεε⎧⎪=++⎪⎪==⎨⎪⎧=⎪=⎨⎪≠⎩⎩subset model ()()1120,,1,2,,,cov ,0,i i i i i j y x E i j n i ji j βεεσεε⎧⎪=+⎪⎪==⎨⎪⎧=⎪=⎨⎪≠⎩⎩(1)Under subset model caculate OLSE 1ˆβfor 1β; (2)Assume full model is true,caculate ()()11ˆˆ,E Var ββ. Attached list:()()()()()0.0250.0250.0250.050.0511 2.201,12 2.1788,5 2.5706,1,11 4.8443,2,12 3.8853t t t F F =====课程编号：MTH17095 北京理工大学2012-2013学年第二学期2010级应用回归分析期末试题A 卷Attached list:()()()0.050.050.041,22 4.30,1,23 4.28,3,22 3.418,F F F ===()()0.0250.02522 2.074,23 2.0687t t ==1.(28)Consider the following model:01122ˆyx x βββ=++,n=25,where y=deliver time (minutes), x 1=number of cases of product, x 2=distance walked by the route driver (feet).Two versions of the model were estimated as follows (the standard errors are in the brackets).(A)12ˆ 2.341 1.6160.014yx x =++ (1.097) (0.171) (0.004)()Re 5784.543,233.732T s SS SS A ==(B) 1ˆ 3.321 2.176yx =+ (1.371) (0.124)()Re 402.134s SS B =(1)Interpret the coefficient of number of cases of product in model (A);(2)Carry out a t-test to test whether for model (A) number of cases of product has a significant effect upon deliver time;()0.05α=(3)Carry out a partial F-test to test whether distance has a significant effect upon deliver time;()0.05α=(4)Test the significance of model(B);()0.05α=(5)Find a 95% confidence interval for the parameter 1β from model (B);(6)Find a 90% Bonferroni confidence interval for the parameter 0β and 1β from model (B); (7)Explain the result in (6).2.(18) Consider the following model:01122ˆyx x βββ=++,n=25,where y=deliver time (minutes), x 1=number of cases of product, x 2=distance walked by the route driver (feet).(1)What are the horizontal scale and vertical scale in the following partial regression plot?What does the plot indicate?(2)It is reported that studentized residual at point 9 9993.2138,0.4983r h ==,where ii h is the ith diagonal element of hat matrix H,and COOK ’s distance 9 3.418D =.Interpret the results. (3)The correlation coefficients 12r between x 1 and x 2 is 120.824r =.What does the result imply? What are sources of the problem?3.(15)To study the relationship between the annual per capita expenditure on education and the annual per capita consumption expenditure,two models are used to fit the data,where y:The annual per capita expenditure on education, x:The annual per capita consumption expenditure.4.(21) Consider the simple linear regression model:011y x ββε=++,with ()()20,E Var εεσ==,and ε uncorrelated.(1)Show ()221R xx E MS S σβ=+; (2) Show ()2Re s E MS σ=.5.(18)A linear regression model is written as follows: 11223344y x x x x ββββε=++++,()()20,E Var εεσ==.The data is shown in the following table:(2)Caculate OLSE 1ˆβ for 1β; (3)Caculate ()1ˆVar β.课程编号：MTH17095 北京理工大学2013—2014学年第二学期2011级应用回归分析期末试题*卷（年份推断为2011，试卷类型未知）附表：()()0.050.0255,10 3.33,10 2.2281F t ==1.(28分)中国民航客运量回归方程为：（括号里是标准误差）12345ˆ450.90.3540.5610.007321.5780.435yx x x x x =+--++, （178.08）（0.085） （0.125） （0.002） （4.030） （0.052）16,13843371.750,13818876.769n SST SSR ===其中：y —民航客运量（万人） x 1—国民收入（亿元） x 2—消费额（亿元）x 3—铁路客运量（万人） x 4—民航航线里程（万公里） x 5—来华旅游入境人数（万人） (1)解释回归方程中民航航线里程的回归系数； (2)检验回归方程的显著性；()0.05α= (3)计算回归方程的决定系数，并作出解释； (4)计算回归的标准误差，解释这一结果； (5)对模型中来华旅游入境人数对民航客运量是否有显著影响进行t-检验； (6)建立x 4的回归系数4β的置信水平为95%的置信区间。

北京理工大学数学专业最优化方法期末试题级A卷级B卷MTH课程编号：MTH17171北京理工大学2014-2015学年第二学期2013级最优化方法期末试题A 卷一、（10分）设()f x 是凸集nS R ?上的凸函数，对12,x x S ∈，实数[]0,1α?，令()121z x x ααα=+-，若z S α∈，证明()()()121f z f x x ααα≥+-。

二、（10分）设数列{}k x 的通项为：22121,2,0,1,!ii i x x x i i +===L ，证明：（1）{}k x 收敛于*0x =；（2）令1,0,1,k k k xx d k +=+=L ，则*lim1k kk x x d →∞-=；（3）{}k x 不是超线性收敛于*x 的。

三、（10分）求解整数规划问题：1212121212min ..14951631,0,,z x x s t x x x x x x x x =-++≤-+≤≥∈Z。

（图解法，割平面法，分枝定界法均可）四、（10分）设f 连续可微有下界，且f ?Lipschitz 连续，即：存在常数0L > ，使得,n x y R ?∈，()()f x f y L x y ?-?≤-，设{}k x 由Wolfe-Powell 型搜索产生，k d 为下降方向，()()cos T k k k kkf xdf x dθ?=-，证明：（1）()220cos kk k f x θ∞=?<∞∑；（2）若0δ?>，使得k ?，cos k θδ≥，则()lim 0k k f x→∞=。

五、（10分）设f 连续可微，序列{}k x 由最速下降法解()min f x ，并做精确搜索产生，证明：0,1,k ?=L ，()()10Tk k f xf x +??=。

六、（10分）已知线性规划：1234123412341234max 2347..23482673,,,0z x x x x s t x x x x x x x x x x x x =++++--=-+-=-≥。

北京理工大学2012-2013学年第一学期2010级《应用随机过程》期末试题A 卷一、（15分）设随机过程()X t Yt Z =+，其中Y ,Z 是相互独立的()0,1N 随机变量，求()X t 的数学期望，协方差函数和一维概率密度函数。

二、（15分）设在(]0,t 内到达某商店的顾客数()X t 是具有强度（每分钟）为λ的泊松过程，求：（1）5分钟内来到的顾客数为2人的概率；（2）5分钟内到来的平均顾客数；（3）设T 为首位顾客到达的时间，计算概率()5P T >。

三、（15分）设质点在线段[]1,5的整数点上作随机游动，n X 表示质点在时刻n 所处的位置，其一步转移概率矩阵为：11000221100022100001110033301000P ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦。

（1）若初始分布为11,,0,0,022⎛⎫ ⎪⎝⎭，求质点在时刻n=1的概率分布； （2）试讨论该Markov 链的状态分类及其各常返闭集的平稳分布。

四、（10分）设Markov 链的状态空间{}0,1,2,I = ，转移概率,10,111,i i i i p p a ---==，1001,1,2,,1i i i a i a ∞-=<<==∑ 。

（1）试证明该Markov 链是不可约常返链； （2）试给出此链正常返的充要条件，并求出状态0的平均返回时间。

五、（15分）某实验室有两台机器，每台机器发生故障的概率为μ，发生故障后立即修理，且在h 时间内机器从故障到正常的概率为()h o h λ+。

令()X t 表示t 时刻正常工作的机器数，则()X t 是一生灭过程。

（1）写出()X t 的Q 矩阵；（2）写出转移概率所满足的Kolmogorov 向前、向后方程；（3）求平稳分布。

六、（15分）设()()cos X t V at =+Θ，其中()0,2,0,1U EV DV πΘ== ，且,V Θ相互独立。

一. (10分) 用三角分解（LU 分解）求解下方程组,要求写出L,U 矩阵：1232644145361182x x x -⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪-= ⎪ ⎪ ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝⎭.二. (10分) 已知矩阵637398785A -⎛⎫ ⎪=- ⎪ ⎪--⎝⎭,求1cond()A 和cond()A ∞,要求计算过程保留三位有效数字,并简要分析所得结果.三. (10分) 设矩阵1001005a A b b a ⎛⎫ ⎪= ⎪ ⎪⎝⎭,且0det()A ≠,试求用,a b 表示的求解线性方程组Ax d =的Jacobi 及Gauss-Seidel 迭代法收敛的充分必要条件.四. (10分) 试确定下求积公式中的待定参数,使求积公式的代数精确度尽量高,并指明所确定的求积公式具有的代数精确度[]20002''()()()()()hh f x dx f f h h f f h α⎡⎤≈++-⎣⎦⎰. 五. (10分) 已知非线性方程240x x +-=在014.x =附近有根，试构造一种收敛的迭代格式,并说明理由.六. (10分) 求形如e (,)bx y a a b =为常数的经验公式,使它能和下表给出的数据相拟合x １ ２ ３ ４ ５ ６ ７ ８y 15.3 20.5 27.4 36.6 49.1 65.6 87.8 117.6七. (10分) 分别用Euler 法和改进Euler 法求解下问题的数值解,取01.h =,计算过程保留四位小数.00201',.,().y x y x y =+≤≤⎧⎨=⎩八. (15分) 用下数据表构造不超过3次的插值多项式,建立导数型插值误差公式,并证明.九. (15分) (1)设''()0,[,]f x x a b <∈, 试证明用梯形公式数值计算积分()baf x dx ⎰所得近似值小于准确值,并说明几何意义.(2)用梯形公式计算⎰,并与真实积分值作比较.。