诚信应考,考试作弊将带来严重后果！
![](https://wkrtcs.bdimg.com/rtcs/image?w=150.67&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_2","w":"150.67","h":"838.67","dataType":"gif","c":"word/media/image1.gif"})
湖南大学课程考试试卷
课程名称： 概率统计A ；课程编码：GE03004 试卷编号： B ；考试时间：120分钟
题 号 | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 | 十 | 总分 |
应得分 | 21 | 51 | 28 | 100 | |||||||
实得分 | |||||||||||
评卷人 | |||||||||||
一、填空题（每小题3分，共21分）
1. 设两两相互独立的三事件A，B和C满足条件： ![](https://wkrtcs.bdimg.com/rtcs/image?w=233.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"233.00","h":"41.00","dataType":"png","c":"word/media/image2_1.png","imgOriW":3242,"imgOriH":574})
![](https://wkrtcs.bdimg.com/rtcs/image?w=184.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"184.00","h":"41.00","dataType":"png","c":"word/media/image3_1.png","imgOriW":2858,"imgOriH":642})
则________________。
2. 设随机变量X服从正态分布![](https://wkrtcs.bdimg.com/rtcs/image?w=115.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"115.00","h":"24.00","dataType":"png","c":"word/media/image5_1.png","imgOriW":2752,"imgOriH":576})
且二次方程无实根的概率为，则_____________。
3. 设随机变量X服从二项分布![](https://wkrtcs.bdimg.com/rtcs/image?w=60.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"60.00","h":"21.00","dataType":"png","c":"word/media/image9_1.png","imgOriW":1440,"imgOriH":512})
，Y服从泊松分布，已知，则X和Y的相关系数________。
4. 设随机变量X和Y的数学期望分别是-2和2，方差分别是1和4，而相关系数为-0.5，根据切比雪夫不等式估计![](https://wkrtcs.bdimg.com/rtcs/image?w=109.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"109.00","h":"27.00","dataType":"png","c":"word/media/image13_1.png","imgOriW":2624,"imgOriH":640})
_____________。
5. 设总体X服从正态分布![](https://wkrtcs.bdimg.com/rtcs/image?w=59.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"59.00","h":"24.00","dataType":"png","c":"word/media/image14_1.png","imgOriW":1408,"imgOriH":576})
，而是来自总体X的简单随机样本，则随机变量所服从的分布为______________。
6. 设![](https://wkrtcs.bdimg.com/rtcs/image?w=93.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"93.00","h":"24.00","dataType":"png","c":"word/media/image17_1.png","imgOriW":2240,"imgOriH":576})
为取自总体的样本，则的矩估计量为______________。
7. 设![](https://wkrtcs.bdimg.com/rtcs/image?w=105.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"105.00","h":"24.00","dataType":"png","c":"word/media/image20_1.png","imgOriW":2528,"imgOriH":576})
为来自正态总体的样本，未知，现要检验假设，则应选取的检验统计量是____________________，当成立时，该统计量服从______________分布。
二、 计算题（共51分）
1. （8分）设随机变量X的密度函数为![](https://wkrtcs.bdimg.com/rtcs/image?w=192.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"192.00","h":"51.00","dataType":"png","c":"word/media/image25_1.png","imgOriW":2657,"imgOriH":701})
，
求：（1）系数A；（2）X的分布函数；（3）![](https://wkrtcs.bdimg.com/rtcs/image?w=69.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"69.00","h":"41.00","dataType":"png","c":"word/media/image26_1.png","imgOriW":1664,"imgOriH":992})
。
2. （8分）设随机变量X服从标准正态分布，求![](https://wkrtcs.bdimg.com/rtcs/image?w=51.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"51.00","h":"26.00","dataType":"png","c":"word/media/image27_1.png","imgOriW":1216,"imgOriH":640})
的概率密度函数。
![](https://wkrtcs.bdimg.com/rtcs/image?w=56.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","w":"56.00","h":"817.33","dataType":"gif","c":"word/media/image28.gif"})
3. （8分）2. 设二维随机变量（X，Y）的联合密度函数为
![](https://wkrtcs.bdimg.com/rtcs/image?w=208.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"208.00","h":"69.00","dataType":"png","c":"word/media/image29_1.png","imgOriW":2388,"imgOriH":796})
，
求：（1）X与Y的边缘密度函数![](https://wkrtcs.bdimg.com/rtcs/image?w=85.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"85.00","h":"24.00","dataType":"png","c":"word/media/image30_1.png","imgOriW":2048,"imgOriH":576})
；（2）X与Y是否相互独立？
4.（9分）设相互独立的两个随机变量X，Y具有同一分布律，且X的分布律为![](https://wkrtcs.bdimg.com/rtcs/image?w=281.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"281.00","h":"41.00","dataType":"png","c":"word/media/image31_1.png","imgOriW":3578,"imgOriH":526})
，求（1）的分布律；（2）的分布律；（3）的分布律。
5.（8分）已知随机变量X和Y分别服从正态分布![](https://wkrtcs.bdimg.com/rtcs/image?w=123.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_19","w":"123.00","h":"24.00","dataType":"png","c":"word/media/image35_1.png","imgOriW":2944,"imgOriH":576})
，且X与Y的相关系数为，设，求（1）Z的数学期望E(Z)与方差D(Z)；（2）X与Z的相关系数。
![](https://wkrtcs.bdimg.com/rtcs/image?w=56.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","w":"56.00","h":"818.67","dataType":"gif","c":"word/media/image39.gif"})
6. （10分）设总体X的概率分布为
![](https://wkrtcs.bdimg.com/rtcs/image?w=128.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_7","w":"128.00","h":"24.00","dataType":"png","c":"word/media/image41_1.png","imgOriW":3072,"imgOriH":576})
，其中是未知参数，利用总体X的如下样本值:
3，1，3，2，3，1，2，3，求![](https://wkrtcs.bdimg.com/rtcs/image?w=13.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_7","w":"13.00","h":"19.00","dataType":"png","c":"word/media/image43_1.png","imgOriW":320,"imgOriH":448})
的矩估计值和最大似然估计值。
三、 应用题（共28分）
1.（10分）1. 甲盒装有1个红球2个黑球，乙盒装有3个红球2个黑球，丙盒装有4个红球1个黑球。现采取掷一骰子决定选盒，出现1、2或3点选甲盒，出现4或5点选乙盒，出现6点选丙盒，在选出的盒里随机摸一球，经过秘密选盒摸球后，宣布摸得一红球，问此球最有可能来自那个盒子？
2. （8分）某种清漆的5个样本，其干燥时间（单位：小时）分别为：6.0，5.8，5.5，6.3，6.4，设干燥时间总体服从正态分布![](https://wkrtcs.bdimg.com/rtcs/image?w=64.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_7","w":"64.00","h":"24.00","dataType":"png","c":"word/media/image44_1.png","imgOriW":1536,"imgOriH":576})
，求的置信度为0.95的置信区间。
3. （10分）食品厂用自动装罐机装罐头食品，每罐标准重量为500克，每隔一定时间需要检验机器的工作情况。现抽25罐，测得其平均值为502克，根据以往的经验，标准差不变。假设重量X服从正态分布![](https://wkrtcs.bdimg.com/rtcs/image?w=73.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_7","w":"73.00","h":"24.00","dataType":"png","c":"word/media/image46_1.png","imgOriW":1760,"imgOriH":576})
，试问该机器工作是否正常？（显著性水平）。
![](https://wkrtcs.bdimg.com/rtcs/image?w=582.00&md5sum=26591b8437ebb9e76e006c73f44ec23d&sign=6820cc1311&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=605&ipr={"t":"img","r":"r_99","w":"582.00","h":"853.00","dataType":"png","c":"word/media/image48.png","imgOriW":638,"imgOriH":932})
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