设随机变量X和Y相互独立,且X~N(0,1),Y的概率分布为P(Y=0)=P(Y=1)=12,则P(X+Y≤12)=(  ) 设随机变量X与Y相互独立,且X服从标准正态分布N(0,1),...

\u8bbe\u968f\u673a\u53d8\u91cfX\u4e0eY\u76f8\u4e92\u72ec\u7acb\uff0c\u4e14X\u670d\u4ece\u533a\u95f4\uff080\uff0c1\uff09\u4e0a\u7684\u5747\u5300\u5206\u5e03\uff0cY\u7684\u6982\u7387\u5206\u5e03\u4e3aP\uff08Y=0\uff09=P\uff08Y=1\uff09=P\uff08Y=2\uff09=1

\u7531\u4e8eX\u7684\u6982\u7387\u5bc6\u5ea6\u4e3afX(x)\uff1d1amp;\uff0c0\uff1cx\uff1c10amp;\uff0c\u5176\u5b83\uff0c\u5206\u5e03\u51fd\u6570\u4e3a\uff1aFX(x)\uff1d0amp;\uff0cx\uff1c0xamp;\uff0c0\u2264x\uff1c11amp;\uff0cx\u22651\u53c8?z\u2208R\uff0cFZ\uff08z\uff09=P\uff08Z\u2264z\uff09=P\uff08XY\u2264z\uff09\u800cY\u662f\u5b9a\u4e49\u4e8e\u540c\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4\u4e4b\u4e0a\u7684\u968f\u673a\u53d8\u6570\u8bbeS=\uff08Y=0\uff09+\uff08Y=1\uff09+\uff08Y=2\uff09\uff0c\u5219\u5229\u7528\u5168\u6982\u7387\u516c\u5f0f\uff0c\u5f97FZ\uff08z\uff09=P\uff08Y=0\uff09P\uff08XY\u2264z|Y=0\uff09+P\uff08Y=1\uff09P\uff08XY\u2264z|Y=1\uff09+P\uff08Y=2\uff09P\uff08XY\u2264z|Y=2\uff09=13P(0\u2264z|Y\uff1d0)+13P(X\u2264z|Y\uff1d1)+13P\uff08X\u2264z|Y=2\uff09=13P(0\u2264z)+13P(X\u2264z)+13P(X\u226412z)\uff08\u5229\u7528X\u4e0eY\u72ec\u7acb\uff09\uff1d13\u00d71+13\u00d7FX(z)+13\u00d7FX(z2)\uff0cz\u2265013\u00d70+13\u00d7FX(z)+13\u00d7FX(z2)\uff0cz\uff1c0\uff1d13+13z+16z\uff0cz\u226500\uff0cz\uff1c0\u2234FZ\uff08z\uff09\u6709\u4e00\u4e2a\u95f4\u65ad\u70b9\uff08z=0\uff09\uff08\u2235limz\u21920+FZ(z)\uff1d13\u2260limz\u21920?FZ(z)\uff1d0\uff09\u6545\u9009\uff1aB

\u7531\u4e8e?z\u2208R\uff0cFZ\uff08z\uff09=P\uff08Z\u2264z\uff09=P\uff08XY\u2264z\uff09\u800cX\uff0cY\u662f\u5b9a\u4e49\u4e8e\u540c\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4\u4e4b\u4e0a\u7684\u968f\u673a\u53d8\u6570\u8bbeS=\uff08Y=0\uff09+\uff08Y=1\uff09\uff0c\u5219\u5229\u7528\u5168\u6982\u7387\u516c\u5f0f\uff0c\u5f97FZ\uff08z\uff09=P\uff08Y=0\uff09P\uff08XY\u2264z|Y=0\uff09+P\uff08Y=1\uff09P\uff08XY\u2264z|Y=1\uff09=12P(0\u2264z|Y\uff1d0)+12P\uff08X\u2264z|Y=1\uff09=12P(0\u2264z)+12P(X\u2264z)\uff08\u5229\u75280\u4e0eY\u72ec\u7acb\uff0cX\u4e0eY\u72ec\u7acb\uff09\uff1d12\u00d71+12\u00d7\u03a6(z)\uff0cz\u2265012\u00d70+12\u00d7\u03a6(z)\uff0cz\uff1c0\uff1d12+12\u03a6(z)\uff0cz\u2265012\u03a6(z)\uff0cz\uff1c0\u2234FZ\uff08z\uff09\u6709\u4e00\u4e2a\u95f4\u65ad\u70b9\uff08z=0\uff09\uff08\u2235limz\u21920+FZ(z)\uff1d12+12\u03a6(0)\uff1d34\u2260limz\u21920?12\u03a6(z)\uff1d14\uff09

因为X~N(0,1),所以


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