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<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62716bd740256a40cedd3866/bg1.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">第六章 主分量分析<span class="_ _0"> </span><span class="ff2 sc1">(PCA- Principle Components Analysis)<span class="_"> </span><span class="ff1">和</span></span></div><div class="t m0 x2 h3 y3 ff1 fs0 fc0 sc0 ls0 ws0">独立分量分析<span class="ff2 sc1"> <span class="_ _0"> </span>( ICA- Independent Components Analysis<span class="fc1">)</span></span></div><div class="t m0 x3 h3 y4 ff2 fs0 fc2 sc1 ls0 ws0">§6-1 <span class="_"> </span><span class="ff1">概述</span> </div><div class="t m0 x4 h4 y5 ff1 fs1 fc3 sc1 ls0 ws0">具有三个特征的两类<span class="_ _1"></span>分类问题，</div><div class="t m0 x5 h4 y6 ff2 fs1 fc3 sc1 ls0 ws0"> <span class="_ _2"></span><span class="ff1">即，特征矢量<span class="_ _3"> </span><span class="ff2 fs0 fc0">{ <span class="ff3">x</span></span></span></div><div class="t m0 x6 h5 y7 ff2 fs2 fc0 sc1 ls0 ws0">1</div><div class="t m0 x7 h6 y6 ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 x8 h5 y7 ff2 fs2 fc0 sc1 ls0 ws0">2</div><div class="t m0 x9 h6 y6 ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 xa h5 y7 ff2 fs2 fc0 sc1 ls0 ws0">3</div><div class="t m0 xb h3 y6 ff2 fs0 fc0 sc1 ls0 ws0"> }<span class="_"> </span><span class="ff1 fc3">：</span></div><div class="t m0 xc h4 y8 ff2 fs1 fc3 sc1 ls0 ws0"> <span class="_ _4"></span><span class="ff1">坐标旋转，即，用新<span class="_ _1"></span>特征<span class="_ _3"> </span><span class="ff2 fs0 fc0">{ <span class="ff3">y</span></span></span></div><div class="t m0 xd h5 y9 ff2 fs2 fc0 sc1 ls0 ws0">1</div><div class="t m0 xe h6 y8 ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">y</span></div><div class="t m0 xf h5 y9 ff2 fs2 fc0 sc1 ls0 ws0">2</div><div class="t m0 x10 h6 y8 ff2 fs0 fc0 sc1 ls0 ws0"> </div><div class="t m0 x4 h4 ya ff2 fs0 fc0 sc1 ls0 ws0">}<span class="_ _0"> </span><span class="ff1 fs1 fc3">代替<span class="_ _5"> </span></span>{ <span class="ff3">x</span></div><div class="t m0 x11 h5 yb ff2 fs2 fc0 sc1 ls0 ws0">1</div><div class="t m0 x12 h6 ya ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 x13 h5 yb ff2 fs2 fc0 sc1 ls0 ws0">2</div><div class="t m0 x14 h6 ya ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 x15 h5 yb ff2 fs2 fc0 sc1 ls0 ws0">3</div><div class="t m0 x16 h4 ya ff2 fs0 fc0 sc1 ls0 ws0"> }<span class="_ _0"> </span><span class="ff1 fs1 fc3">得到：</span></div><div class="t m0 x4 h4 yc ff1 fs1 fc3 sc1 ls0 ws0">再次坐标旋转，可见<span class="_ _1"></span>，只用一个</div><div class="t m0 x17 h4 yd ff2 fs1 fc3 sc1 ls0 ws0"> <span class="_ _6"></span><span class="ff1">特征<span class="_ _3"> </span><span class="ff2 fs0 fc0">{<span class="ff3">z</span>}<span class="_ _0"> </span></span>，就足以分开。</span></div><div class="t m0 x4 h4 ye ff1 fs1 fc3 sc1 ls0 ws0">模式识别的空间，从</div><div class="t m0 x18 h6 yf ff2 fs0 fc0 sc1 ls0 ws0">{ <span class="ff3">x</span></div><div class="t m0 x19 h5 y10 ff2 fs2 fc0 sc1 ls0 ws0">1</div><div class="t m0 x17 h6 yf ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 x1a h5 y10 ff2 fs2 fc0 sc1 ls0 ws0">2</div><div class="t m0 x1b h6 yf ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">x</span></div><div class="t m0 x12 h5 y10 ff2 fs2 fc0 sc1 ls0 ws0">3</div><div class="t m0 x1c h6 yf ff2 fs0 fc0 sc1 ls0 ws0"> } <span class="_ _1"></span><span class="fc3">=> </span>{ <span class="ff3">y</span></div><div class="t m0 x1d h5 y10 ff2 fs2 fc0 sc1 ls0 ws0">1</div><div class="t m0 x1e h6 yf ff2 fs0 fc0 sc1 ls0 ws0">, <span class="ff3">y</span></div><div class="t m0 x1f h5 y10 ff2 fs2 fc0 sc1 ls0 ws0">2</div><div class="t m0 xa h6 yf ff2 fs0 fc0 sc1 ls0 ws0"> }<span class="_ _7"></span><span class="fc3">=></span>{<span class="ff3">z</span>}</div><div class="t m0 x20 h4 y11 ff2 fs1 fc3 sc1 ls0 ws0">“<span class="_ _8"></span><span class="ff1">称为<span class="_ _3"> </span><span class="fc0">降维<span class="_ _1"></span></span><span class="ff2">”</span>。<span class="_ _9"> </span><span class="ff2">“<span class="_ _a"></span><span class="ff1">可降维的前提是，原<span class="_ _1"></span>来的特征空间具有<span class="_ _3"> </span><span class="fc0">冗余<span class="_ _1"></span></span><span class="ff2">”</span>。</span></span></span></div></div></div><div class="pi" data-data='{"ctm":[1.333333,0.000000,0.000000,1.333333,0.000000,0.000000]}'></div></div>
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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62716bd740256a40cedd3866/bg2.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x21 h3 y12 ff2 fs0 fc0 sc1 ls0 ws0">§6-2<span class="_"> </span><span class="ff1">随机变量的统计特性基础</span></div><div class="t m0 x21 h4 y13 ff1 fs1 fc3 sc1 ls0 ws0">一、随机变量与随机矢<span class="_ _1"></span>量的概率分布</div><div class="t m0 x4 h7 y14 ff4 fs1 fc3 sc1 ls0 ws0"></div><div class="t m0 x22 h8 y15 ff5 fs1 fc3 sc2 ls0 ws0">概率密度分布<span class="_ _5"> </span><span class="ff6 sc1">:</span></div><div class="t m0 x4 h6 y16 ff1 fs3 fc3 sc1 ls0 ws0"><span class="fc4 sc1">设有</span><span class="ff2 fs1"><span class="fc4 sc1"> </span><span class="_"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">M</span></span><span class="fc4 sc1"> </span><span class="_ _b"> </span><span class="fs3"><span class="fc4 sc1"> </span><span class="_ _c"></span><span class="ff1"><span class="fc4 sc1">维特征矢量</span><span class="_ _d"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span><span class="fs1 fc3"><span class="fc4 sc1"> </span><span class="_"> </span></span></span><span class="fc4 sc1">，</span></span></span></span></div><div class="t m0 x23 h9 y17 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">1)</span></div><div class="t m0 x3 ha y18 ff5 fs1 fc3 sc2 ls0 ws0">假设 <span class="_ _5"> </span><span class="ff3 fs4 fc0 sc1">x<span class="fs5">m</span><span class="fc2"> <span class="_ _e"> </span></span></span>是连续变量，则其<span class="_ _1"></span>概率可以表示为：</div><div class="t m0 x3 ha y19 ff5 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">第</span><span class="_ _5"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">m</span><span class="_"> </span></span><span class="sc2"><span class="fc4 sc1">行是</span><span class="fc4 sc1">特征</span><span class="fc4 sc1"> </span><span class="_ _5"> </span></span><span class="ff3 fs4 fc0"><span class="fc4 sc1">x</span></span></div><div class="t m0 x24 hb y1a ff3 fs6 fc0 sc1 ls0 ws0"><span class="fc4 sc1">m</span></div><div class="t m0 x25 h8 y19 ff5 fs1 fc3 sc2 ls0 ws0"><span class="fc4 sc1">的第</span><span class="fc4 sc1"> </span><span class="_ _5"> </span><span class="ff3 fs0 fc0 sc1"><span class="fc4 sc1">n</span><span class="fs1 fc2"><span class="fc4 sc1"> </span><span class="_"> </span></span></span><span class="fc4 sc1">次采</span><span class="fc4 sc1">样值</span><span class="fc4 sc1">，第 </span><span class="_ _5"> </span><span class="ff3 fs0 fc0 sc1"><span class="fc4 sc1">n</span></span><span class="ff2 sc1"><span class="fc4 sc1"> </span><span class="_"> </span></span><span class="fc4 sc1">列是所</span><span class="fc4 sc1">有特</span><span class="fc4 sc1">征的第</span><span class="fc4 sc1"> </span><span class="_ _5"> </span><span class="ff3 fs0 fc0 sc1"><span class="fc4 sc1">n</span></span><span class="ff2 sc1"><span class="fc4 sc1"> </span><span class="_"> </span></span><span class="fc4 sc1">次采</span></div><div class="t m0 x3 h8 y1b ff5 fs1 fc3 sc2 ls0 ws0">样值。</div><div class="t m0 x26 h9 y1c ff2 fs1 fc3 sc1 ls0 ws0">(6-2)</div><div class="t m0 x27 h9 y1d ff2 fs1 fc3 sc1 ls0 ws0">(6-3)</div><div class="t m0 x28 h9 y1e ff2 fs1 fc3 sc1 ls0 ws0">(6-4)</div><div class="t m0 x29 h6 y16 ff2 fs3 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _c"></span><span class="ff1"><span class="fc4 sc1">对特征矢量</span><span class="_ _d"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span><span class="fs3"><span class="fc4 sc1"> </span><span class="_ _f"> </span><span class="fc3"><span class="fc4 sc1"> </span><span class="_ _10"></span><span class="ff1"><span class="fc4 sc1">进行多次采样，得到</span><span class="_ _d"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">N</span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_"> </span></span><span class="fc4 sc1">个样本：</span></span></span></span></span></span></div><div class="t m0 x3 h6 y1f ff2 fs0 fc0 sc1 ls0 ws0"><span class="fc4 sc1">X</span></div><div class="t m0 x2a h5 y20 ff3 fs2 fc0 sc1 ls0 ws0"><span class="fc4 sc1">n</span></div><div class="t m0 x2b h4 y1f ff2 fs1 fc0 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _11"> </span><span class="fc3"><span class="fc4 sc1"> </span><span class="_ _12"></span><span class="ff1"><span class="fc4 sc1">是特</span><span class="fc4 sc1">征矢</span><span class="fc4 sc1">量</span><span class="_ _3"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _13"> </span><span class="fc4 sc1"> </span><span class="_ _6"></span><span class="ff1"><span class="fc4 sc1">的第</span><span class="_ _3"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">n</span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _14"> </span><span class="fc4 sc1"> </span><span class="_ _15"></span><span class="ff1"><span class="fc4 sc1">次</span><span class="fc4 sc1">采样</span><span class="fc4 sc1">结果，</span><span class="fc4 sc1">即，</span><span class="_ _1"></span><span class="fc4 sc1">随机矢</span><span class="fc4 sc1">量</span><span class="_ _3"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _13"> </span><span class="fc4 sc1"> </span><span class="_ _6"></span><span class="ff1"><span class="fc4 sc1">的第</span><span class="_ _3"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">n</span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _16"> </span></span><span class="fc4 sc1">个</span><span class="fc4 sc1">实现</span><span class="fc4 sc1">。</span></span></span></span></span></span></span></span></span></div></div></div><div class="pi" data-data='{"ctm":[1.333333,0.000000,0.000000,1.333333,0.000000,0.000000]}'></div></div>
<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62716bd740256a40cedd3866/bg3.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x3 ha y21 ff3 fs4 fc3 sc1 ls0 ws0">x<span class="fs5">m<span class="_"> </span><span class="ff1 fs1">取某个指定值的概率为<span class="_ _5"> </span><span class="ff2">0 <span class="_"> </span></span>，即：</span></span></div><div class="t m0 x3 h4 y22 ff1 fs1 fc3 sc1 ls0 ws0">但在某区间上取值不<span class="_ _1"></span>为<span class="_ _16"> </span><span class="ff2">0<span class="_ _1"></span> <span class="_ _16"> </span></span>：</div><div class="t m0 x3 h4 y23 ff1 fs1 fc3 sc2 ls0 ws0"><span class="fc4 sc1">正态</span><span class="fc4 sc1">分布</span><span class="fc4 sc1">：</span></div><div class="t m0 x4 h8 y24 ff5 fs1 fc3 sc2 ls0 ws0"><span class="fc4 sc1">随机</span><span class="fc4 sc1">矢量</span><span class="fc4 sc1">的概</span><span class="fc4 sc1">率：</span></div><div class="t m0 x27 h9 y25 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">4</span><span class="fc4 sc1">)</span></div><div class="t m0 x2c h9 y26 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">5)</span></div><div class="t m0 x2d hc y27 ff2 fs7 fc1 sc1 ls0 ws0">X</div><div class="t m1 x2e hd y28 ff7 fs8 fc1 sc1 ls0 ws0">0</div><div class="t m2 x2d he y29 ff2 fs9 fc1 sc1 ls0 ws0">X</div><div class="t m0 x2f hf y2a ff1 fsa fc1 sc1 ls0 ws0">表示</div><div class="t m0 x30 h4 y2b ff2 fs1 fc0 sc1 ls0 ws0"><span class="fc4 sc1">X</span><span class="_"> </span><span class="ff1 fc3"><span class="fc4 sc1">中的</span><span class="fc4 sc1">每个分</span><span class="fc4 sc1">量都</span><span class="fc4 sc1">小于</span><span class="fc4 sc1">或等于</span><span class="_ _5"> </span></span><span class="fc4 sc1">X</span><span class="fs5"><span class="fc4 sc1">0</span><span class="_"> </span></span><span class="ff1 fc3"><span class="fc4 sc1">中相应</span><span class="fc4 sc1">分量</span><span class="_ _1"></span><span class="fc4 sc1">，即</span></span></div><div class="t m0 x31 h10 y2c ff1 fsa fc1 sc1 ls0 ws0">中的相应分量<span class="_ _17"></span>，即<span class="fsb"> </span></div><div class="t m0 x4 h4 y2d ff1 fs1 fc3 sc2 ls0 ws0">其中：</div><div class="t m0 x28 h9 y2e ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">6)</span></div><div class="t m0 x32 h4 y11 ff1 fs1 fc3 sc1 ls0 ws0">也就是多个随机变量<span class="_ _1"></span>的联合分布。</div><div class="t m0 x33 h7 y18 ff4 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"></span></div><div class="t m0 xe h4 y2f ff1 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">随机</span><span class="fc4 sc1">矢量的</span><span class="fc4 sc1">概率</span><span class="fc4 sc1">密度函</span><span class="_ _1"></span><span class="fc4 sc1">数</span><span class="_ _16"> </span><span class="ff2"><span class="fc4 sc1">:</span></span></div></div></div><div class="pi" data-data='{"ctm":[1.333333,0.000000,0.000000,1.333333,0.000000,0.000000]}'></div></div>
<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62716bd740256a40cedd3866/bg4.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x34 h11 y30 ff2 fsc fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _18"></span><span class="ff1"><span class="fc4 sc1">联合分布的</span><span class="_ _19"> </span><span class="ff3 fsd fc0"><span class="fc4 sc1">x</span></span><span class="sc2"><span class="fc4 sc1"> </span><span class="_ _1a"> </span></span><span class="fc4 sc1">边缘分布分别</span></span></div><div class="t m0 x35 h4 y31 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _2"></span><span class="ff1"><span class="fc4 sc1">两个</span><span class="fc4 sc1">随机</span><span class="fc4 sc1">变量</span><span class="_ _3"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">x</span><span class="fs1 fc2"><span class="fc4 sc1"> </span><span class="_ _1b"> </span></span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _1c"></span><span class="ff1"><span class="fc4 sc1">和</span><span class="_ _3"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">y</span><span class="fc2"><span class="fc4 sc1"> </span><span class="_ _1d"> </span></span></span><span class="ff2"><span class="fc4 sc1"> </span><span class="_ _2"></span><span class="ff1"><span class="fc4 sc1">，</span><span class="fc4 sc1">或随</span><span class="fc4 sc1">机矢量</span><span class="_ _3"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span><span class="fc4 sc1">=</span><span class="fc4 sc1">[</span><span class="ff3"><span class="fc4 sc1">x</span></span><span class="fc4 sc1">, </span><span class="fc4 sc1"> </span><span class="ff3"><span class="fc4 sc1">y</span></span><span class="fc4 sc1">]</span><span class="_ _1d"> </span></span><span class="ff2"> <span class="_ _2"></span><span class="ff1">，其<span class="_ _1"></span>联合分布<span class="_ _3"> </span><span class="ff3 fs0 fc0">p<span class="ff2">(</span>x<span class="ff2">, </span>y<span class="ff2">)</span></span></span></span></span></span></span></span></span></div><div class="t m0 x36 h9 y1c ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span>6-11<span class="_ _1"></span>)</div><div class="t m0 x37 h4 y32 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="_ _1"></span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="_ _1e"> </span><span class="fc4 sc1"> </span><span class="_ _1f"></span><span class="ff1"><span class="fc4 sc1">给定</span><span class="_ _20"> </span><span class="fc4 sc1">，即</span><span class="_ _3"> </span><span class="ff2 fs0 fc0"><span class="fc4 sc1">X</span></span></span></div><div class="t m0 x38 h5 y33 ff2 fs2 fc0 sc1 ls0 ws0"><span class="fc4 sc1">0</span></div><div class="t m0 x39 h6 y32 ff2 fs0 fc0 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="fc4 sc1">≤</span><span class="fc4 sc1"> </span><span class="fc4 sc1">X</span><span class="fc4 sc1"> </span><span class="fc4 sc1">≤</span><span class="fc4 sc1"> </span><span class="fc4 sc1">X</span></div><div class="t m0 x3a h5 y33 ff2 fs2 fc0 sc1 ls0 ws0"><span class="fc4 sc1">1</span></div><div class="t m0 x3b h4 y32 ff2 fs0 fc0 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _0"> </span><span class="ff1 fs1 fc3"><span class="fc4 sc1">的</span><span class="fc4 sc1">概率</span><span class="fc4 sc1">为：</span></span></div><div class="t m0 x3c h4 y34 ff2 fs1 fc3 sc1 ls0 ws0"> <span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="fc4 sc1"> </span><span class="_ _1e"> </span> <span class="_ _2"></span><span class="ff1">给定<span class="_ _1e"> </span><span class="fc4 sc1">，</span><span class="_ _1"></span>即<span class="_ _21"> </span><span class="ff2 fs0 fc0">X</span></span></div><div class="t m0 x3d h5 y35 ff2 fs2 fc0 sc1 ls0 ws0">0</div><div class="t m0 x3e h4 y34 ff2 fs0 fc0 sc1 ls0 ws0"> ≤ X <span class="_ _0"> </span><span class="ff1 fs1 fc3">的概率为：</span></div><div class="t m0 x3 h4 y36 ff2 fs1 fc3 sc1 ls0 ws0"> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _22"> </span><span class="ff1">即，<span class="_ _5"> </span><span class="ff3 fs0 fc0">y<span class="_"> </span></span>取任意值情况下，<span class="_ _16"> </span><span class="ff3 fs0 fc0">x<span class="_"> </span></span>的分布。实际上<span class="_ _1"></span>，</span></div><div class="t m0 x12 h4 y37 ff2 fs1 fc3 sc1 ls0 ws0"> <span class="_ _12"></span><span class="ff1">就是不理睬<span class="_ _21"> </span><span class="ff3 fs0 fc0">y<span class="_ _1"></span> <span class="_ _23"> </span></span><span class="ff2"> <span class="_ _12"></span><span class="ff1">而单独观察<span class="_ _21"> </span><span class="ff3 fs0 fc0">x</span><span class="sc2"> <span class="_ _24"> </span></span><span class="ff2"> <span class="_ _25"></span><span class="ff1">时，它的分布情况。同<span class="_ _1"></span>理可以定义<span class="_ _21"> </span><span class="ff3 fs0 fc0">y <span class="_"> </span></span>的边</span></span></span></span></span></div><div class="t m0 x3 h4 y38 ff1 fs1 fc3 sc1 ls0 ws0">沿分布<span class="_ _5"> </span><span class="ff8">:</span></div></div></div><div class="pi" data-data='{"ctm":[1.333333,0.000000,0.000000,1.333333,0.000000,0.000000]}'></div></div>
<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62716bd740256a40cedd3866/bg5.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x3 h3 y39 ff1 fs0 fc3 sc2 ls0 ws0">二、相关矩阵和协方差矩阵</div><div class="t m0 x39 h3 y3a ff1 fs1 fc3 sc2 ls0 ws0">随机矢量的数学期望<span class="_ _1"></span><span class="fs0">：</span></div><div class="t m0 x3f h4 y3b ff1 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">称为</span><span class="fc4 sc1">随机</span><span class="fc4 sc1">矢量</span><span class="ff2"><span class="fc4 sc1"> </span><span class="_"> </span><span class="fc2"><span class="fc4 sc1">X</span></span><span class="fs0"><span class="fc4 sc1"> </span><span class="_"> </span></span></span><span class="fc4 sc1">的均方</span><span class="fc4 sc1">差。</span></div><div class="t m0 x40 h4 y3c ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _26"></span><span class="ff1"><span class="fc4 sc1">随机</span><span class="fc4 sc1">矢量</span><span class="_ _21"> </span><span class="ff2 fc2"><span class="fc4 sc1">X</span><span class="_ _1"></span><span class="fc4 sc1"> </span><span class="_"> </span></span><span class="fc4 sc1">的</span><span class="fc4 sc1">方差</span><span class="fc4 sc1">的估计</span><span class="fc4 sc1">：</span></span></div><div class="t m0 x3 h4 y3d ff1 fs1 fc3 sc1 ls0 ws0">设<span class="ff2"> <span class="_"> </span><span class="ff3 fc2">g<span class="ff2">(X)<span class="_"> </span></span></span></span>是随机矢量<span class="ff2"> <span class="_"> </span><span class="fc2">X</span> <span class="_"> </span></span>的导出量，则<span class="ff2"> <span class="_"> </span><span class="ff3 fc2">g<span class="ff2">(X) <span class="_"> </span></span></span></span>的数学期望定义为<span class="fc1">：</span></div><div class="t m0 x41 h9 y3e ff2 fs1 fc3 sc1 ls0 ws0">(6-12<span class="_ _1"></span>)</div><div class="t m0 x4 h4 y3f ff1 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">当</span><span class="ff2"><span class="fc4 sc1"> </span><span class="_"> </span><span class="ff3 fc2"><span class="fc4 sc1">g</span><span class="ff2"><span class="fc4 sc1">(</span><span class="fc4 sc1">X</span><span class="fc4 sc1">)</span><span class="fc4 sc1">=X</span><span class="fc4 sc1"> </span><span class="_"> </span></span></span></span><span class="fc4 sc1">时得</span><span class="fc4 sc1">到</span></div><div class="t m0 x41 h9 y40 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">1</span><span class="fc4 sc1">3</span><span class="_ _1"></span><span class="fc4 sc1">)</span></div><div class="t m0 x37 h4 y41 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_ _6"></span><span class="ff1"><span class="fc4 sc1">如果</span><span class="_ _21"> </span><span class="ff3 fs0 fc0"><span class="fc4 sc1">p</span></span></span></div><div class="t m0 x42 h5 y42 ff2 fs2 fc0 sc1 ls0 ws0"><span class="fc4 sc1">X</span></div><div class="t m0 x34 h4 y41 ff2 fs0 fc0 sc1 ls0 ws0"><span class="fc4 sc1">(X</span><span class="fc4 sc1">)</span><span class="fs1 fc3"><span class="fc4 sc1"> </span><span class="_"> </span><span class="ff1"><span class="fc4 sc1">为对</span><span class="fc4 sc1">称分</span><span class="fc4 sc1">布，</span></span></span></div><div class="t m0 x3 h4 y43 ff1 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">则，</span><span class="_ _5"> </span><span class="ff2 fc0"><span class="fc4 sc1">m</span></span></div><div class="t m0 x42 h12 y44 ff2 fse fc0 sc1 ls0 ws0"><span class="fc4 sc1">X</span></div><div class="t m0 x1a h4 y43 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1"> </span><span class="_"> </span><span class="ff1"><span class="fc4 sc1">的估计</span><span class="fc4 sc1">为</span><span class="_ _16"> </span></span><span class="fc4 sc1">:</span></div><div class="t m0 x2c h4 y17 ff1 fs1 fc3 sc2 ls0 ws0"><span class="fc4 sc1">当</span><span class="_ _27"> </span><span class="sc1"><span class="fc4 sc1">时</span><span class="_ _16"> </span><span class="ff3 fc2"><span class="fc4 sc1">g</span><span class="_ _1"></span><span class="ff2"><span class="fc4 sc1">(</span><span class="fc4 sc1">X</span><span class="fc4 sc1">)</span><span class="fc4 sc1"> </span><span class="_"> </span></span></span><span class="fc4 sc1">的</span><span class="fc4 sc1">数学期</span><span class="fc4 sc1">望：</span></span></div><div class="t m0 x43 h9 y45 ff2 fs1 fc3 sc1 ls0 ws0"><span class="fc4 sc1">(</span><span class="fc4 sc1">6</span><span class="fc4 sc1">-</span><span class="fc4 sc1">1</span><span class="fc4 sc1">7</span><span class="_ _1"></span><span class="fc4 sc1">)</span></div><div class="t m0 x44 h3 y46 ff1 fs0 fc3 sc1 ls0 ws0">例如，轮盘赌，获得<span class="_ _0"> </span><span class="ff8 fc0">41<span class="_ _0"> </span><span class="ff1">元</span></span>奖金的概率是<span class="_ _0"> </span><span class="ff8 fc0">1/<span class="_ _17"></span>51<span class="_ _0"> </span><span class="ff1 fc3">，<span class="_ _0"> </span></span>15<span class="_ _0"> </span><span class="ff1">元<span class="fc3">奖金的概</span></span></span></div><div class="t m0 x44 h3 y47 ff1 fs0 fc3 sc1 ls0 ws0">率<span class="_ _0"> </span><span class="ff8 fc0">3/51<span class="_ _0"> </span></span>，<span class="_ _0"> </span><span class="fc0 sc0">8<span class="_ _28"> </span></span>元奖金的概率<span class="_ _0"> </span><span class="ff8 fc0">5/51<span class="_ _0"> </span></span>，<span class="_ _0"> </span><span class="fc0 sc0">5<span class="_ _28"> </span><span class="sc1">元</span></span>奖金的概率<span class="_ _0"> </span><span class="ff8 fc0">7/<span class="_ _17"></span>51<span class="_ _0"> </span><span class="ff1 fc3">，<span class="_ _0"> </span><span class="fc0 sc0">3<span class="_ _28"> </span><span class="sc1">元</span></span></span></span></div><div class="t m0 x44 h3 y48 ff1 fs0 fc3 sc1 ls0 ws0">奖金的概率<span class="_ _0"> </span><span class="ff8 fc0">10/51<span class="_ _0"> </span></span>，<span class="_ _0"> </span><span class="fc0 sc0">2<span class="_ _28"> </span><span class="sc1">元</span></span>奖金的概率<span class="_ _0"> </span><span class="ff8 fc0">25/51<span class="_ _0"> </span></span>，如果在每个位置</div><div class="t m0 x44 h3 y49 ff1 fs0 fc3 sc1 ls0 ws0">都下注一元，则每把必赢，赢钱的期望值为：</div><div class="t m0 x1 h13 y4a ff8 fs0 fc3 sc1 ls0 ws0">41<span class="ff2 fc0">/51+<span class="fc3">15<span class="fc5">×</span></span>3/51+<span class="fc3">8<span class="fc5">×</span></span>5/5<span class="_ _1"></span>1+<span class="fc3">5<span class="fc5">×</span></span>7/51+<span class="fc3">3<span class="fc5">×</span></span>10/51+<span class="fc3">2<span class="fc5">×</span></span>25/51 = 4.7255 </span></div><div class="t m0 x45 h6 y4b ff2 fs0 fc0 sc1 ls0 ws0">4.7255 <span class="fc3">– 6</span>= <span class="fc3">– </span>1.2745</div><div class="t m0 x44 h3 y4c ff1 fs0 fc3 sc1 ls0 ws0">只在<span class="_ _0"> </span><span class="ff2 fc0">15<span class="_ _0"> </span><span class="ff1">元、<span class="_ _0"> </span></span>2<span class="_ _0"> </span><span class="ff1">元</span></span>处个押<span class="fc0">一元，</span>则期望值为<span class="fc0">：</span></div><div class="t m0 x46 h6 y4d ff2 fs0 fc3 sc1 ls0 ws0">15<span class="fc5">×<span class="fc0">3/51+</span></span>2<span class="fc5">×<span class="fc0">25/51</span> <span class="fc0">= 1.8627 1.8627 </span></span>– 2<span class="fc0"> = </span>– <span class="fc0">0.1373</span></div></div></div><div class="pi" data-data='{"ctm":[1.333333,0.000000,0.000000,1.333333,0.000000,0.000000]}'></div></div>
