1 . 在信息理论中,
和
是两个取值相同的离散型随机变量,分布列分别为:
,
,
,
,
,
.定义随机变量
的信息量
,
和
的“距离”
.
(1)若
,求
;
(2)已知发报台发出信号为0和1,接收台收到信号只有0和1.现发报台发出信号为0的概率为
,由于通信信号受到干扰,发出信号0接收台收到信号为0的概率为
,发出信号1接收台收到信号为1的概率为
.
(ⅰ)若接收台收到信号为0,求发报台发出信号为0的概率;(用
,
表示结果)
(ⅱ)记随机变量
和
分别为发出信号和收到信号,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b08fcbcf19c6ca72cd66c201ef43f9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4380cd57f824c5d9df1ca493cbd8cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfe82ce73937d36166659f21492c825e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1a870945a04cd86f2e0026fc53a2b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0e3b00fe47801afb53ec56706c21a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27b4e8e7a49dbe86419e00672d1927c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd67429e1b0f56bc66a547fc9c6eed2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5633fa4fa8837dff506561b7943715fb.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17d0c830d39efe08dad4f2104325b8c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59a8bb9552579e3cd3c7d693ce37b445.png)
(2)已知发报台发出信号为0和1,接收台收到信号只有0和1.现发报台发出信号为0的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29c8578f06897aa6fb84aa95c797d3d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d9b426bcc34a2cca2184dc1310f5e4.png)
(ⅰ)若接收台收到信号为0,求发报台发出信号为0的概率;(用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
(ⅱ)记随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3719852c05eef71dd595791e3dc10de7.png)
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2卷引用:海南省部分学校2024届高三考前押题考试(三模)数学试题
2 . 设椭圆
,定义椭圆
的“相关圆”方程为
.若抛物线
的焦点与椭圆
的一个焦点重合,且椭圆
短轴 一个端点和其两个焦点构成直角三角形.
(Ⅰ)求椭圆
的方程和“相关圆”
的方程;
(Ⅱ)过“相关圆”
上任意一点
的直线
与椭圆
交于
两点.
为坐标原点,若
,证明原点
到直线
的距离是定值,并求
的取值范围.
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/56afc5a624634baf9baec213d168d965.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/253e88c11236431280145d6cd6f000e0.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/5cbb63c62d644ec59ac7538f9d249f48.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/5fb669fe466d43fc9ee205915cfd633f.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/253e88c11236431280145d6cd6f000e0.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/253e88c11236431280145d6cd6f000e0.png)
(Ⅰ)求椭圆
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/253e88c11236431280145d6cd6f000e0.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/82e21fdb8cbd4260812a54b696ca9905.png)
(Ⅱ)过“相关圆”
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/82e21fdb8cbd4260812a54b696ca9905.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/4db8d7e2e1a64e3a8ef3d2a042910840.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/04d81cc80e484800a627346c0162ad65.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/253e88c11236431280145d6cd6f000e0.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/3da383c4cc1f4c889e47c4827692f911.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/5f44075e9e7c4610a775eee1b87d3643.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/497082593a4d49dd88246073e271c763.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/5f44075e9e7c4610a775eee1b87d3643.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/1aaa2cfc28e0428192ecec5dfd149b88.png)
![](https://img.xkw.com/dksih/QBM/2016/7/22/1572927734513664/1572927740076032/STEM/c774ef38bea9455ca68a9093db57a162.png)
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