某超市计划按月订购一种酸奶，每天进货量相同，进货成本每瓶4元，售价每瓶6元，未售出的酸奶降价处理，以每瓶2元的价格当天全部处理完．根据往年销售经验，每天需求量与当天最高气温（单位：℃）有关．如果最高气温不低于25，需求量为500瓶；如果最高气温位于区间[20，25），需求量为300瓶；如果最高气温低于20，需求量为200瓶．为了确定六月份的订购计划，统计了前三年六月份各天的最高气温数据，得下面的频数分布表：
最高气温 | [10，15） | [15，20） | [20，25） | [25，30） | [30，35） | [35，40） |
天数 | 2 | 16 | 36 | 25 | 7 | 4 |
以最高气温位于各区间的频率代替最高气温位于该区间的概率。
（1）求六月份这种酸奶一天的需求量X（单位：瓶）的分布列；
（2）设六月份一天销售这种酸奶的利润为Y（单位：元），当六月份这种酸奶一天的进货量n（单位：瓶）为多少时，Y的数学期望达到最大值？
本小题主要考查统计等基础知识， 考查分析问题能力和运算求解能力.
（1）由题意知，![](https://wkrtcs.bdimg.com/rtcs/image?w=17.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_7","w":"17.33","h":"16.00","dataType":"gif","c":"word/media/image1.gif"})
所有的可能取值为200,300,500，由表格数据知
![](https://wkrtcs.bdimg.com/rtcs/image?w=154.67&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"154.67","h":"37.33","dataType":"gif","c":"word/media/image2.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=134.67&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"134.67","h":"37.33","dataType":"gif","c":"word/media/image3.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=176.00&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"176.00","h":"37.33","dataType":"gif","c":"word/media/image4.gif"})
.
因此的分布列为
|
|
|
|
| 0.2 | 0.4 | 0.4 |
![](https://wkrtcs.bdimg.com/rtcs/image?w=16.00&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_7","w":"16.00","h":"16.00","dataType":"gif","c":"word/media/image9.gif"})
由题意知，这种酸奶一天的需求量至多为500，至少为200，因此只需考虑
当![](https://wkrtcs.bdimg.com/rtcs/image?w=89.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_7","w":"89.33","h":"17.33","dataType":"gif","c":"word/media/image11.gif"})
时，
若最高气温不低于25，则Y=6n-4n=2n
若最高气温位于区间![](https://wkrtcs.bdimg.com/rtcs/image?w=62.67&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_7","w":"62.67","h":"22.67","dataType":"gif","c":"word/media/image12.gif"})
，则Y=6×300+2（n-300）-4n=1200-2n;
若最高气温低于20，则Y=6×200+2（n-200）-4n=800-2n;
因此EY=2n×0.4+（1200-2n）×0.4+(800-2n) ×0.2=640-0.4n
当![](https://wkrtcs.bdimg.com/rtcs/image?w=85.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_7","w":"85.33","h":"17.33","dataType":"gif","c":"word/media/image13.gif"})
时，
若最高气温不低于20，则Y=6n-4n=2n;
若最高气温低于20，则Y=6×200+2（n-200）-4n=800-2n;
因此EY=2n×(0.4+0.4)+(800-2n)×0.2=160+1.2n
所以n=300时，Y的数学期望达到最大值，最大值为520元。
本文来源：https://www.2haoxitong.net/k/doc/157d80bcf80f76c66137ee06eff9aef8941e48a6.html
文档为doc格式
![](https://wkrtcs.bdimg.com/rtcs/image?w=17.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"17.33","h":"16.00","dataType":"gif","c":"word/media/image1.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=26.67&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"26.67","h":"17.33","dataType":"gif","c":"word/media/image5.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=25.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"25.33","h":"17.33","dataType":"gif","c":"word/media/image6.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=25.33&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"25.33","h":"17.33","dataType":"gif","c":"word/media/image7.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=14.67&md5sum=153af6410e14a0d4471e5b14e7a49d61&sign=60700839ff&rtcs_flag=2&rtcs_ver=3&l=webapp&bucketNum=101&ipr={"t":"img","r":"r_6","w":"14.67","h":"16.00","dataType":"gif","c":"word/media/image8.gif"})