多元统计分析A卷11

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广西工学院2023—2023学年第2学期课程考核试题

考核课程应用多元统计分析（A卷）考核班级统计091，092学生数80印数90考核方式闭卷考核时间120分钟

一、（10分）证明随机向量X协方差矩阵?为非负定矩阵.

二、（15分）设随机向量X的期望向量为E(X)?(5,?2,7)?，协方差矩阵

?412???V(X)??19?3?，

?2?325???Y1?X1?X2?X3,Y2?X1?X2,Y3?X1?2X2，求Y?(Y1,Y2,Y3)?的数学期望与协方差矩阵.三、（10分）设X，Y是随机向量，A,B为常数矩阵，证明：cov(AX,BY)?Acov(X,Y)B?.四、（10分）设随机向量X的协方差矩阵为

?16?43???V(X)???44??2

?3?29???求其相关系数矩阵R.

?1?21?五、（10分）设X~N3(?,?),A???，其中

??110??16?42?????11?1?,????44?1??，

?2?14???求Y?AX的分布.

六、（15分）已知两个总体?1，?2的概率密度分别为f1(x)和f2(x)，且总体的先验概率分布为p1?0.7,p2?0.3，误判损失为c(2|1)?200,c(1|2)?500.（1）建立Bayes判别准则；

（2）设有一新样品x0满足f1(x0)?6.3,f2(x0)?1，判定x0的归属问题.七、（15分）设有10个样本?i(i?1,2,?,10)的距离矩阵为：

第1页共4页

???8?9??1?2?3?4?5?6?710???1?0???2?20???3?210???4?7560??D??5?64550?

??6?666970???7?6659720???8?66597110????9?7761085340???10?98889101010100?用最短距离法对这10样品进行聚类，并画出树形图。八、（15分）下表是某城市在42天中午12点的空气数据.HC太阳辐射风速（x1）CO（x3）NO（x4）NO2（x5）O3（x6）（x2）（x7）8.007.007.0010.006.008.009.005.007.008.006.006.007.0010.0010.009.008.008.009.009.0010.009.008.005.006.008.006.0098.00107.00103.0088.0091.0090.0084.0072.0082.0064.0071.0091.0072.0070.0072.0077.0076.0071.0067.0069.0062.0088.0080.0030.0083.0084.0078.007.004.004.005.004.005.007.006.005.005.005.004.007.004.004.004.004.005.004.003.005.004.004.003.005.003.004.002.003.003.002.002.002.004.004.001.002.004.002.004.002.001.001.001.003.002.003.003.002.002.003.001.002.002.0012.009.005.008.008.0012.0012.0021.0011.0013.0010.0012.0018.0011.008.009.007.0016.0013.009.0014.007.0013.005.0010.007.0011.008.005.006.0015.0010.0012.0015.0014.0011.009.003.007.0010.007.0010.0010.007.004.002.005.004.006.0011.002.0023.006.0011.002.003.003.004.003.004.005.004.003.004.003.003.003.003.003.003.003.004.003.003.004.003.004.003.004.003.003.00第2页共4页

8.0079.002.001.007.0010.003.006.0062.004.003.009.008.003.0010.0037.003.001.007.002.003.008.0071.004.001.0010.007.003.007.0052.004.001.0012.008.004.005.0048.006.005.008.004.003.006.0075.004.001.0010.0024.003.0010.0035.004.001.006.009.002.008.0085.004.001.009.0010.002.005.0086.003.001.006.0012.002.005.0086.007.002.0013.0018.002.007.0079.007.004.009.0025.003.007.0079.005.002.008.006.002.006.0068.006.002.0011.0014.003.008.0040.004.003.006.005.002.00利用相关系数矩阵对数据进行主成分分析及因子分析得到的结果如下：说明的总方差初始特征值方差的3.38319.80017.20230.3879.3357.6672.227提取平方和载入合计方差的%累积%2.33733.38333.3831.38619.80053.1831.20417.20230.384成分1234567合计2.3371.3861.204.727.653.537.156累积3.38353.18370.38480.77190.10697.773100.000提取方法：主成分分析。

旋转成分矩阵a成分2-.174.736.274-.383.151.820.071风速太阳辐射CONONO2O3HC1-.027.043.705.644.811.166.7063.840-.017-.391-.483.002-.152.467提取方法:主成分分析法。旋转法:具有Kaise