概率论与数理统计 设随机变量X~e(2),Y~P(3)则E(X+Y＾2）怎么算 概率论与数理统计 求方差问题 D(X+Y)怎么算？

\u8bbe\u968f\u673a\u53d8\u91cfX~e(2) Y~e(4)\uff0c\u6c42E(X+Y),E(2X-3Y^2)

X~e(2) Y~e(4)
\u2234E(X)=1/2 E(Y)=1/4 D(X)=1/4 D(Y)=1/16
E(X+Y)=E(X)+EY=3/4
E(2X-3Y²)=2E(X)-3E(Y²)
D(Y)+(EY)²=E(Y²)=1/8
E(2X-3Y²)=1-3/8=5/8

\u7531X~N\uff080\uff0c4\uff09\u4e0eY~N\uff082\uff0c3/4\uff09\u4e3a\u6b63\u6001\u5206\u5e03\u5f97\uff1a
X~N\uff080\uff0c4\uff09\u6570\u5b66\u671f\u671bE\uff08X\uff09\uff1d0\uff0c\u65b9\u5deeD\uff08X\uff09\uff1d4\uff1b
Y~N\uff082\uff0c3/4\uff09\u6570\u5b66\u671f\u671bE\uff08Y\uff09\uff1d2\uff0c\u65b9\u5deeD\uff08Y\uff09\uff1d4/3\u3002
\u7531X\uff0cY\u76f8\u4e92\u72ec\u7acb\u5f97\uff1a
E\uff08XY\uff09\uff1dE\uff08X\uff09E\uff08Y\uff09\uff1d0\u00d72\uff1d0\uff0c
D\uff08X\uff0bY\uff09\uff1dD\uff08X\uff09\uff0bD\uff08Y\uff09\uff1d4\u00d74/3\uff1d16/3\uff0c
D\uff082X\uff0d3Y\uff09\uff1d2²D\uff08X\uff09\uff0d3²D\uff08Y\uff09\uff1d4\u00d74\uff0d9\u00d74/3\uff1d4

\u6269\u5c55\u8d44\u6599 :
1. \u6b63\u6001\u5206\u5e03\u6027\u8d28\uff1a
\u2474 \u4e00\u822c\u6b63\u6001\u5206\u5e03\u8bb0\u4e3aX~N\uff08\u03bc\uff0c\u03c3²\uff09\uff0c\u6807\u51c6\u6b63\u6001\u5206\u5e03\u8bb0\u4e3aX~N\uff080\uff0c1\uff09\u3002
\u2475 \u4e00\u822c\u6b63\u6001\u5206\u5e03\u8f6c\u5316\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03:\u82e5X~N\uff08\u03bc\uff0c\u03c3²\uff09\uff0cY\uff1d\uff08X\uff0d\u03bc\uff09/\u03c3 ~N\uff080\uff0c1\uff09\u3002
\u2476 \u6b63\u6001\u5206\u5e03\u6570\u5b66\u671f\u671b\u4e3aE\uff08X\uff09\uff1d\u03bc\uff0cD\uff08X\uff09\uff1d\u03c3²\u3002
2. \u6570\u5b66\u671f\u671b\u4e0e\u65b9\u5dee\u6027\u8d28:
\u8bbeC\u4e3a\u4e00\u4e2a\u5e38\u6570\uff0cX\u548cY\u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\u2474 \u6570\u5b66\u671f\u671b\u6027\u8d28\uff1a
E\uff08C\uff09\uff1dC\uff0cE\uff08CX\uff09\uff1dCE\uff08X\uff09\uff0cE\uff08X\uff0bY\uff09\uff1dE\uff08X\uff09\uff0bE\uff08Y\uff09\uff0c\u5728X\u548cY\u76f8\u4e92\u72ec\u7acb\u65f6\u6709E\uff08XY\uff09\uff1dE\uff08X\uff09E\uff08Y\uff09\u3002
\u2475\u65b9\u5dee\u6027\u8d28\uff1a
D\uff08C\uff09\uff1d0\uff0cD\uff08CX\uff09\uff1dC²D\uff08X\uff09\uff0cD\uff08X\uff0bC\uff09\uff1dD\uff08X\uff09\uff0c\u5728X\u548cY\u76f8\u4e92\u72ec\u7acb\u65f6\u6709D\uff08X\uff0bY\uff09\uff1dD\uff08X\uff09\uff0bD\uff08Y\uff09\u3002
\u53c2\u8003\u8d44\u6599 :
\u767e\u5ea6\u767e\u79d1_\u6570\u5b66\u671f\u671b
\u767e\u5ea6\u767e\u79d1_\u6b63\u6001\u5206\u5e03
\u767e\u5ea6\u767e\u79d1_\u65b9\u5dee

如图