概率论与数理统计 设随机变量X~N(0,1)求,D(X^2)

\u6982\u7387\u8bba\u4e0e\u6570\u7406\u7edf\u8ba1 \u8bbe\u968f\u673a\u53d8\u91cfX~N\uff080\uff0c1\uff09\u6c42\uff0cE\uff08X^2\uff09

D\uff08x\uff09=E\uff08X^2\uff09-[E\uff08X\uff09]^2=1
E\uff08X^2\uff09=1

\u8bbe\u968f\u673a\u53d8\u91cfX\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3af(x)=A/x^2\uff0cx>100\uff1b0\uff0cx<=100\uff0c\u7cfb\u6570A\u4e3a10\u3002
A\uff1d1\uff0f\uff08\u222b\uff3b\uff0d\u221e\uff0c\uff0b\u221e\uff3df\uff08x\uff09dx\uff09
\uff1d1\uff0f\uff08\u222b\uff3b10\uff0c\uff0b\u221e\uff3da\uff0fx\uff3e2dx\uff09
\uff1d1\uff0f\uff08\uff0da\uff0fx\uff5c\uff3b10\uff0c\uff0b\u221e\uff3d\uff09
\uff1d1\uff0f\uff08a\uff0f10\uff09

\uff1d10
\u6269\u5c55\u8d44\u6599\uff1a
\u5982\u679c\u968f\u673a\u53d8\u91cf\u53ea\u53d6\u5f97\u6709\u9650\u4e2a\u503c\u6216\u65e0\u7a77\u80fd\u6309\u4e00\u5b9a\u6b21\u5e8f\u4e00\u4e00\u5217\u51fa\uff0c\u5176\u503c\u57df\u4e3a\u4e00\u4e2a\u6216\u82e5\u5e72\u4e2a\u6709\u9650\u6216\u65e0\u9650\u533a\u95f4\uff0c\u8fd9\u6837\u7684\u968f\u673a\u53d8\u91cf\u5c5e\u4e8e\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u3002\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7531\u968f\u673a\u53d8\u91cf\u53d6\u503c\u8303\u56f4\u786e\u5b9a\u3002\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u4e00\u5207\u53ef\u80fd\u7684\u53d6\u503c\u4e0e\u5bf9\u5e94\u7684\u6982\u7387\u4e58\u79ef\u4e4b\u548c\u4e3a\u5176\u6570\u5b66\u671f\u671b\u7684\u52a0\u6743\u5e73\u5747\u3002
\u53d8\u91cf\u53d6\u503c\u53ea\u80fd\u53d6\u79bb\u6563\u578b\u7684\u4efb\u610f\u6709\u9650\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u81ea\u7136\u6570\uff0c\u5982\u679c\u53d8\u91cf\u53ef\u4ee5\u5728\u67d0\u4e2a\u533a\u95f4\u5185\u53d6\u4efb\u4e00\u5b9e\u6570\uff0c\u5219\u53d8\u91cf\u7684\u53d6\u503c\u53ef\u4ee5\u662f\u8fde\u7eed\u7684\u3002

D(X^2)=E(X^4)-(E(X^2))^2
E(X^2)=D(X)+(E(X))^2
E(X^4)就得积分了，用两次分部积分就可以了，写起来比较麻烦就不写了

X^2~X²(1) 卡方分布

D(X^2)=2

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