{"id":228,"date":"2016-08-19T18:53:40","date_gmt":"2016-08-19T10:53:40","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=228"},"modified":"2016-12-07T05:12:43","modified_gmt":"2016-12-07T05:12:43","slug":"guan-yu-zheng-tai-fen-bu-de-fan-li","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=228","title":{"rendered":"\u5173\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u53cd\u4f8b"},"content":{"rendered":"<p>\u6982\u7387\u8bba\u4e2d\u4e00\u4e2a\u57fa\u672c\u7684\u5b9a\u7406\u662f\u8bf4<br \/>\n<div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 1<\/span><span class='latex_thm_h'>.<\/span> \u5982\u679c$(X,Y)$\u670d\u4ece\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03, \u5373\u5176\u5bc6\u5ea6\u51fd\u6570\u662f:<br \/>\n$$<br \/>\nf(x,y)=\\frac{1}{2\\pi\\sigma_X\\sigma_Y\\sqrt{1-\\rho^2}}\\exp\\Bigg(-\\frac{1}{2(1-\\rho^2)}\\Big[<br \/>\n\\frac{(x-\\mu_X)^2}{\\sigma_X^2}+<br \/>\n\\frac{(y-\\mu_Y)^2}{\\sigma_Y^2}<br \/>\n-\\frac{2\\rho(x-\\mu_X)(y-\\mu_Y)}{\\sigma_X\\sigma_Y}<br \/>\n\\Big]\\Bigg)<br \/>\n$$<br \/>\n\u5219<br \/>\n<ul><li>$aX+bY$\u670d\u4ece\u4e00\u7ef4\u6b63\u6001\u5206\u5e03;<\/li><li>$X,Y$\u662f\u72ec\u7acb\u7684\u5f53\u4e14\u4ec5\u5f53$X,Y$\u662f\u4e0d\u76f8\u5173\u7684.<\/li><\/ul><\/div><br \/>\n<!--more--><br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 1<\/span><span class='latex_rem_h'>.<\/span> \u82e5\u4ee4,<br \/>\n$$<br \/>\n\\mathbf{\\mu}=\\pmatrix{\\mu_X\\\\\\mu_Y},\\quad<br \/>\n\\mathbf{\\Sigma}=\\pmatrix{\\sigma_X^2&amp;\\rho\\sigma_X\\sigma_Y\\\\\\rho\\sigma_X\\sigma_Y&amp;\\sigma_Y^2},<br \/>\n$$<br \/>\n\u5219\u4e0a\u8ff0\u516c\u5f0f\u662f$k=2$\u65f6, \u4e0b\u5217\u516c\u5f0f\u7684\u7279\u4f8b,<br \/>\n$$<br \/>\nf_{\\mathbf{x}}(x_1,\\ldots,x_k)=\\frac{1}{\\sqrt{(2\\pi)^k|\\Sigma|}}\\exp\\Big(<br \/>\n-\\frac{1}{2}(\\mathbf{x}-\\mathbf{\\mu})^T\\mathbf{\\Sigma}^{-1}(\\mathbf{x}-\\mathbf{\\mu})<br \/>\n\\Big).<br \/>\n$$<br \/>\n<\/div><br \/>\n\u4e00\u4e2a\u81ea\u7136\u7684\u53cd\u95ee\u9898\u662f<br \/>\n<div class='latex_prob'><span class='latex_prob_h'>\u95ee\u9898 1<\/span><span class='latex_prob_h'>.<\/span> \u82e5$X,Y$\u5206\u522b\u662f1\u7ef4\u6b63\u6001\u5206\u5e03, \u4e14\u5b83\u4eec\u662f\u4e0d\u76f8\u5173\u7684, \u90a3\u4e48<br \/>\n<ul><li>$(X,Y)$\u662f\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u5417?<\/li><li>\u6b64\u65f6$X,Y$\u662f\u72ec\u7acb\u7684\u5417?<\/li><\/ul><\/div><\/p>\n<p>\u56de\u5fc6, $X$\u662f\u4e00\u7ef4\u6b63\u6001\u5206\u5e03\u662f\u6307, \u5176\u5bc6\u5ea6\u51fd\u6570\u6ee1\u8db3<br \/>\n$$<br \/>\nf_X(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma_X}\\exp\\big(-\\frac{(X-\\mu_X)^2}{2\\sigma_X^2}\\big).<br \/>\n$$<br \/>\n\u800c\u4e24\u4e2a\u968f\u673a\u53d8\u91cf$X,Y$\u4e0d\u76f8\u5173\u5b9a\u4e49\u4e3a\u5b83\u4eec\u7684\u76f8\u5173\u7cfb\u6570\u4e3a0, \u5176\u4e2d\u76f8\u5173\u7cfb\u6570\u5b9a\u4e49\u4e3a<br \/>\n$$<br \/>\nr=\\frac{E(XY)-E(X)E(Y)}{\\sqrt{D(X)D(Y)}},<br \/>\n$$<br \/>\n\u5176\u4e2d$D(X)=E((X-E(X))^2)$\u662f$X$\u7684\u65b9\u5dee.<\/p>\n<p>\u4e0b\u9762\u7684\u4f8b\u5b50\u5c06\u8868\u660e, \u4e0a\u9762\u7684\u95ee\u9898\u7684\u7b54\u6848\u90fd\u662f\u5426\u5b9a\u7684.<br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 1<\/span><span class='latex_examp_h'>.<\/span> \u5047\u8bbe$X\\sim N(0,1)$\u5373\u670d\u4ece\u671f\u671b$\\mu_X=0$, \u65b9\u5dee$\\sigma_X=1$\u7684\u6807\u51c6\u6b63\u6001\u5206\u5e03. \u800c$W$\u662fRademacher\u5206\u5e03, \u5373$W=-1$\u6216\u8005$1$, \u5176\u6982\u7387\u90fd\u4e3a$1\/2$. \u4e14\u5047\u8bbe$W$\u4e0e$X$\u662f\u72ec\u7acb\u7684. \u4ee4$Y=WX$, \u5219<br \/>\n<ul><li>$X,Y$\u662f\u4e0d\u76f8\u5173\u7684;<\/li><li>$X,Y$\u90fd\u670d\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03;<\/li><li>$X,Y$<span class=\"latex_em\">\u4e0d\u662f<\/span>\u72ec\u7acb\u7684.<\/li><\/ul><\/div><br \/>\n<div class='latex_rem'><span class='latex_rem_h'>\u6ce8\u8bb0 2<\/span><span class='latex_rem_h'>.<\/span> \u8fd9\u4e2a\u4f8b\u5b50\u6765\u6e90\u4e8eWIKI: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Normally_distributed_and_uncorrelated_does_not_imply_independent\" target=\"_blank\">Normally distributed and uncorrelated does not imply independent<\/a>.<br \/>\n<\/div><\/p>\n<p>\u9996\u5148\u6765\u770b$X,Y$\u662f\u4e0d\u76f8\u5173. \u4e8b\u5b9e\u4e0a<br \/>\n\\begin{align*}<br \/>\nE(XY)-E(X)E(Y)&amp;=E(XY)-E(X)E(WX)\\\\<br \/>\n&amp;=E(XY)-E(X)E(W)E(X)\\quad\\text{($W,X$\u72ec\u7acb)}\\\\<br \/>\n&amp;=E(XY)-0\\cdot E(X)^2\\quad\\text{($E(W)=0$)}\\\\<br \/>\n&amp;=E(E(XY|W))\\quad\\text{($E(E(x|y))=E(x))$,\u8fd9\u5e94\u8be5\u53eb\u5168\u6982\u7387\u516c\u5f0f)}\\\\<br \/>\n&amp;=P(W=1)E(XY|W=1)+P(W=-1)E(XY|W=-1)\\\\<br \/>\n&amp;=\\frac{1}{2}\\left(E(X^2|W=1)+E(-X^2|W=-1)\\right)\\\\<br \/>\n&amp;=\\frac{1}{2}\\left(E(X^2)+E(-X^2)\\right)\\\\<br \/>\n&amp;=\\frac{1}{2}\\left(E(X^2)-E(X^2)\\right)\\\\<br \/>\n&amp;=0.<br \/>\n\\end{align*}<\/p>\n<p>\u63a5\u4e0b\u6765, \u8bc1\u660e$Y$\u4e5f\u670d\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03. \u4e8b\u5b9e\u4e0a\u5229\u7528\u5168\u6982\u7387\u516c\u5f0f\u5f97\u5230<br \/>\n\\begin{align*}<br \/>\nP(Y\\leq s)&amp;=P(WX\\leq s)=P(WX\\leq s|W=1)P(W=1)+P(WX\\leq s|W=-1)P(W=-1)\\\\<br \/>\n&amp;=\\frac{1}{2}\\left(P(X\\leq s)+P(-X\\leq s)\\right)\\\\<br \/>\n&amp;=P(X\\leq s),<br \/>\n\\end{align*}<br \/>\n\u8fd9\u91cc\u6211\u4eec\u7528\u5230\u4e86$X$\u670d\u4ece\u6807\u51c6\u6b63\u6001\u5206\u5e03, \u5176\u5bc6\u5ea6\u51fd\u6570\u5173\u4e8e$\\mu=0$\u5bf9\u79f0, \u4ece\u800c$P(X\\leq s)=P(-X\\leq s)$.<\/p>\n<p>\u6700\u540e, \u4e3a\u4e86\u8bf4\u660e$X,Y$\u4e0d\u72ec\u7acb, \u53ea\u9700\u6ce8\u610f\u5230, \u5bf9\u535a\u5185\u5c14\u51fd\u6570$f(t)=|t|$, \u6211\u4eec\u6709$f(X)=|X|=|Y|=f(Y)$. \u800c\u6982\u7387\u8bba\u4e2d\u4e00\u4e2a\u7ed3\u8bba\u8868\u660e, \u82e5$X,Y$\u662f\u72ec\u7acb\u7684, \u5219\u5bf9\u4efb\u610f\u7684\u535a\u5185\u5c14\u51fd\u6570$f$, \u90fd\u6709$f(X),f(Y)$\u662f\u72ec\u7acb\u7684. \u6ce8\u610f, \u6309\u7167\u72ec\u7acb\u7684\u5b9a\u4e49\u53ef\u77e5, \u4e00\u4e2a\u4e8b\u4ef6$A$\u548c\u81ea\u5df1\u662f\u72ec\u7acb\u7684, \u5f53\u4e14\u4ec5\u5f53<br \/>\n$$<br \/>\nP(A\\cap A)=P(A)\\cdot P(A),<br \/>\n$$<br \/>\n\u8fd9\u8868\u660e$P(A)=0$\u6216\u8005$1$. \u800c\u5bf9\u4e0a\u9762\u7684\u60c5\u5f62, \u82e5\u5b9a\u4e49$A=\\left\\{X:|X|&lt;s\\right\\}$, \u5219$P(A)=P(|X|&lt;s)\\in(0,1)$ (\u5bf9\u6240\u6709\u7684\u5b9e\u6570$s$). <\/p>\n<p>\u5f53\u7136, \u53e6\u4e00\u4e2a\u72ec\u7acb\u7684\u76f4\u63a5\u63a8\u8bba\u662f\u6761\u4ef6\u6982\u7387\u6ee1\u8db3 $$ P(A|B)=P(A\\cap B)\/P(B)=P(A)P(B)\/P(B)=P(A). $$ \u5bf9\u4e0a\u9762\u7684\u60c5\u5f62, \u76f4\u63a5\u8003\u5bdf$A=\\left\\{Y:Y&gt;1\\right\\}$, $B=\\left\\{X:X=1\/2\\right\\}$, \u5219<br \/>\n$$<br \/>\nP(A|B)=P(Y&gt;1|X=1\/2)=P(X&gt;1|X=1\/2)=0,<br \/>\n$$<br \/>\n\u8fd9\u91cc\u6211\u4eec\u7528\u5230\u4e86$Y$\u4e0e$X$\u670d\u4ece\u76f8\u540c\u7684\u5206\u5e03. \u4e0a\u5f0f\u8868\u660e<br \/>\n$$<br \/>\n0=P(A|B)\\neq P(A)=P(Y&gt;1)&gt;0.<br \/>\n$$<\/p>\n<p>\u6211\u4eec\u77e5\u9053, \u4f5c\u4e3a\u82e5$(X,Y)$\u7684\u8054\u5408\u5206\u5e03\u662f\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03, \u5219\u5176\u4efb\u610f\u7ebf\u6027\u7ec4\u5408$aX+bY$\u4e5f\u662f\u4e00\u7ef4\u6b63\u6001\u5206\u5e03(\u5176\u5b9e\u53cd\u8fc7\u6765\u4e5f\u5bf9). \u6211\u4eec\u5c06\u7528\u8fd9\u4e00\u4e8b\u5b9e\u8bf4\u660e, \u4e0a\u9762\u4f8b\u5b50\u7ed9\u51fa\u7684$X,Y$\u7684\u8054\u5408\u5206\u5e03\u4e0d\u662f2\u7ef4\u6b63\u6001\u5206\u5e03. \u4e8b\u5b9e\u4e0a, \u6ce8\u610f\u5230<br \/>\n\\begin{align*}<br \/>\nP(X+Y=0)&amp;=P(X+Y=0|W=-1)P(W=-1)\\\\<br \/>\n&#038;\\qquad+P(X+Y=0|W=1)P(W=1)\\\\<br \/>\n&amp;=\\frac{1}{2}\\left(P(0=0|W=-1)+P(2X=0|W=1)\\right)\\\\<br \/>\n&amp;=\\frac{1}{2}(1+0)=\\frac{1}{2}.<br \/>\n\\end{align*}<br \/>\n\u53ef\u89c1$X+Y$\u4e0d\u53ef\u80fd\u662f\u6b63\u6001\u5206\u5e03(\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u53d6\u5f97\u67d0\u7ed9\u5b9a\u503c\u7684\u6982\u7387\u4e3a0).<\/p>\n<p>\u5f53\u7136, \u4e5f\u5bb9\u6613\u8bc1\u660e, $(X,Y)$\u670d\u4ece\u4e8c\u7ef4\u6b63\u6001\u65f6, $X,Y$\u72ec\u7acb\u5f53\u4e14\u4ec5\u5f53$X,Y$\u4e0d\u76f8\u5173. \u8fd9\u4e5f\u53ef\u4ee5\u8bf4\u660e\u4e0a\u9762\u4f8b\u5b50\u4e2d\u7684$(X,Y)$\u7684\u8054\u5408\u5206\u5e03\u4e0d\u662f2\u7ef4\u6b63\u6001\u5206\u5e03.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u7387\u8bba\u4e2d\u4e00\u4e2a\u57fa\u672c\u7684\u5b9a\u7406\u662f\u8bf4 \u5b9a\u7406 1. \u5982\u679c$(X,Y)$\u670d\u4ece\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03, \u5373\u5176\u5bc6\u5ea6\u51fd\u6570\u662f: $$ f(x,&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=228\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u5173\u4e8e\u6b63\u6001\u5206\u5e03\u7684\u53cd\u4f8b<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[23,40,42],"class_list":["post-228","post","type-post","status-publish","format-standard","hentry","category-math","tag-bu-xiang-guan","tag-zheng-tai-fen-bu","tag-du-li","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/228","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=228"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/228\/revisions"}],"predecessor-version":[{"id":274,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/228\/revisions\/274"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=228"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=228"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}