设随机变量X与Y相互独立,且X~N(1,4²),Y~N(1,1²) 设随机变量X与Y相互独立,且X~N(0,1),N(1,1)则...
\u8bbe\u968f\u673a\u53d8\u91cfx\u548cy\u76f8\u4e92\u72ec\u7acb\u4e14X~N(1,2),Y~N(2,2)\uff0c\u52192X+3Y\u670d\u4ece\u5206\u5e03= \u8981\u5177\u4f53\u6b65\u9aa4\u8bb0\u4f4f\u6b63\u6001\u5206\u5e03\u7684\u57fa\u672c\u516c\u5f0f
\u5982\u679c\u4e24\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u6b63\u6001\u5206\u5e03
X~N(\u03bc1, m²)\uff0cY~N(\u03bc2\uff0cn²)
\u90a3\u4e48Z=aX\u00b1bY\u4ecd\u7136\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c
Z~N(a\u03bc1\u00b1b\u03bc2\uff0ca²m²+b²n²)
\u4e8e\u662f\u8fd9\u91cc\u76842X+3Y~N(2*1+3*2\uff0c4*2+9*2)
\u5373\u670d\u4ece\u5206\u5e03N(8\uff0c26)
X,Y\u90fd\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u90a3\u4e48X+Y\u4e5f\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u4e14X+Y~N(1,2)\uff0c\u8868\u793ax+y\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7684\u5bf9\u79f0\u8f74\u662f1\u3002
\u90a3\u4e48p(X+Y\u5c0f\u4e8e\u7b49\u4e8e1)=1/2
\u76f8\u5f53\u4e8e\u6574\u4e2a\u51fd\u6570\u4e0e\u5750\u6807\u8f74\u56f4\u6210\u9762\u79ef\u7684\u5de6\u534a\u90e8\u5206\u4e3a0.5\u3002
\u968f\u673a\u53d8\u91cf\u7684\u53d6\u503c\u843d\u5728\u67d0\u4e2a\u533a\u57df\u4e4b\u5185\u7684\u6982\u7387\u5219\u4e3a\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5728\u8fd9\u4e2a\u533a\u57df\u4e0a\u7684\u79ef\u5206\u3002\u5f53\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5b58\u5728\u7684\u65f6\u5019\uff0c\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7684\u79ef\u5206\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u7531\u4e8e\u968f\u673a\u53d8\u91cfX\u7684\u53d6\u503c \u53ea\u53d6\u51b3\u4e8e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u7684\u79ef\u5206\uff0c\u6240\u4ee5\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u5728\u4e2a\u522b\u70b9\u4e0a\u7684\u53d6\u503c\u5e76\u4e0d\u4f1a\u5f71\u54cd\u968f\u673a\u53d8\u91cf\u7684\u8868\u73b0\u3002
\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u548cX\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u53d6\u503c\u4e0d\u540c\u7684\u70b9\u53ea\u6709\u6709\u9650\u4e2a\u3001\u53ef\u6570\u65e0\u9650\u4e2a\u6216\u8005\u76f8\u5bf9\u4e8e\u6574\u4e2a\u5b9e\u6570\u8f74\u6765\u8bf4\u6d4b\u5ea6\u4e3a0\uff08\u662f\u4e00\u4e2a\u96f6\u6d4b\u96c6\uff09\uff0c\u90a3\u4e48\u8fd9\u4e2a\u51fd\u6570\u4e5f\u53ef\u4ee5\u662fX\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3002
\u8fde\u7eed\u578b\u7684\u968f\u673a\u53d8\u91cf\u53d6\u503c\u5728\u4efb\u610f\u4e00\u70b9\u7684\u6982\u7387\u90fd\u662f0\u3002\u4f5c\u4e3a\u63a8\u8bba\uff0c\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u5728\u533a\u95f4\u4e0a\u53d6\u503c\u7684\u6982\u7387\u4e0e\u8fd9\u4e2a\u533a\u95f4\u662f\u5f00\u533a\u95f4\u8fd8\u662f\u95ed\u533a\u95f4\u65e0\u5173\u3002\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6982\u7387P{x=a}=0\uff0c\u4f46{X=a}\u5e76\u4e0d\u662f\u4e0d\u53ef\u80fd\u4e8b\u4ef6\u3002
本题中，E(X)=1,D(X)=4²、E(Y)=1,D(Y)=1，∴μ=E(Z)=E(X-3Y+2)=E(X)-3E(Y)+2=0，δ²=D(Z)=D(X-3Y+2)=D(X)+9D(Y)=16+9=5²，
∴Z~N(0,5²)。
供参考。
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