當(dāng)前位置：首頁(yè) > 编程资源 > 编程问答 >内容正文

编程问答

依分布收敛、依概率收敛、均方收敛、几乎处处收敛

發(fā)布時(shí)間：2023/12/20 编程问答 24 豆豆

生活随笔收集整理的這篇文章主要介紹了依分布收敛、依概率收敛、均方收敛、几乎处处收敛小編覺(jué)得挺不錯(cuò)的,現(xiàn)在分享給大家,幫大家做個(gè)參考.

1. 分布函數(shù)列的弱收斂性

弱收斂 ${F_n(x)\}$ 是分布函數(shù)列，如果存在一個(gè)函數(shù) $F_(x)$ , 對(duì)任意 $x$ , 使得 $lim?n→∞Fn(x)=F(x)\lim_{n \to \infty}F_n(x)=F(x)$ , 則稱(chēng) $F_n(x)$ 弱收斂于 $F (x)$ , 并記為
$Fn(x)→wF(x).F_n(x)~\overset{w}{\underset{}{\to}}~F(x).$

2. 隨機(jī)變量的四種收斂性

假定如下的隨機(jī)變量 $X$ 與隨機(jī)變量序列 ${X_n\}$ 定義在概率空間 $(Ω,F,P)(\Omega,F,P)$ 上.

依分布收斂 設(shè) $X_n$ 的分布函數(shù)為 $F_n(x)$ , $X$ 的分布函數(shù)為 $F (x)$ , 假如
$Fn(x)→wF(x)，F_n(x)\overset{w}{\underset{}{\to}}F(x)，$
則稱(chēng) ${X_n\}$ 依分布收斂于 $X$ , 記為 $Xn→dX.X_n~\oversetozvdkddzhkzd{\underset{}{\to}}~X.$

依分布收斂為這里討論收斂性中最弱的收斂，其只能保證分布函數(shù)序列收斂。

依概率收斂 若對(duì) $?\forall$ $ε>0\varepsilon>0$ ,
$lim?n→∞P{w:∣Xn(w)?X(w)∣≥ε}=0,\lim_{n \to \infty}P\{w: |X_n(w)-X(w)| \geq \varepsilon\}=0,$
則稱(chēng) ${X_n\}$ 依概率收斂于 $X$ , 記為 $Xn→PX.X_n~\overset{P}{\underset{}{\to}}~X.$

依概率收斂強(qiáng)于依分布收斂，其先給定 $ε>0\varepsilon>0$ ，保證 $∣Xn(w)?X(w)∣≥ε|X_n(w)-X(w)| \geq \varepsilon$ 隨著 $n$ 增大，概率趨向于0。這里可見(jiàn) $ε\varepsilon$ 越小收斂性越好。

r-階收斂 假設(shè) $E∣X∣r<∞E|X|^r<\infty$ , $E∣Xn∣r<∞,r>0E|X_n|^r<\infty, r>0$ . 如果
$lim?n→∞E∣Xn?X∣r=0，\lim_{n \to \infty}E|X_n-X|^r=0，$
則稱(chēng) ${X_n\}$ r-階收斂于 $X$ 。其中當(dāng) $r = 2$ 時(shí)，稱(chēng)為均方收斂。

r-階收斂強(qiáng)于依概率收斂，其說(shuō)明隨著 $n$ 的增大，r-階的 $X_n-X|$ 的期望是為0的。

幾乎處處收斂（依概率1收斂） 如果
$P(w:lim?n→∞Xn(w)=X(w))=1，P(w:\lim_{n \to \infty}X_n(w)=X(w))=1，$
稱(chēng) ${X_n\}$ 幾乎處處收斂于 $X$ , 記為 $Xn→a.s.X.X_n~\overset{a.s.}{\underset{}{\to}}~X.$

幾乎處處收斂也強(qiáng)于依概率收斂，和依概率收斂比較，可以看成取定 $ε=0\varepsilon=0$ ，所以也叫依概率1收斂。事實(shí)上，幾乎處處收斂是不收斂點(diǎn)的集合是只構(gòu)成勒貝格零集。

3. 收斂性間關(guān)系

#mermaid-svg-aeBu7OBWqCKz38Ds .label{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);fill:#333;color:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .label text{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .node rect,#mermaid-svg-aeBu7OBWqCKz38Ds .node circle,#mermaid-svg-aeBu7OBWqCKz38Ds .node ellipse,#mermaid-svg-aeBu7OBWqCKz38Ds .node polygon,#mermaid-svg-aeBu7OBWqCKz38Ds .node path{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-aeBu7OBWqCKz38Ds .node .label{text-align:center;fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .node.clickable{cursor:pointer}#mermaid-svg-aeBu7OBWqCKz38Ds .arrowheadPath{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .edgePath .path{stroke:#333;stroke-width:1.5px}#mermaid-svg-aeBu7OBWqCKz38Ds .flowchart-link{stroke:#333;fill:none}#mermaid-svg-aeBu7OBWqCKz38Ds .edgeLabel{background-color:#e8e8e8;text-align:center}#mermaid-svg-aeBu7OBWqCKz38Ds .edgeLabel rect{opacity:0.9}#mermaid-svg-aeBu7OBWqCKz38Ds .edgeLabel span{color:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .cluster rect{fill:#ffffde;stroke:#aa3;stroke-width:1px}#mermaid-svg-aeBu7OBWqCKz38Ds .cluster text{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:12px;background:#ffffde;border:1px solid #aa3;border-radius:2px;pointer-events:none;z-index:100}#mermaid-svg-aeBu7OBWqCKz38Ds .actor{stroke:#ccf;fill:#ECECFF}#mermaid-svg-aeBu7OBWqCKz38Ds text.actor>tspan{fill:#000;stroke:none}#mermaid-svg-aeBu7OBWqCKz38Ds .actor-line{stroke:grey}#mermaid-svg-aeBu7OBWqCKz38Ds .messageLine0{stroke-width:1.5;stroke-dasharray:none;stroke:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .messageLine1{stroke-width:1.5;stroke-dasharray:2, 2;stroke:#333}#mermaid-svg-aeBu7OBWqCKz38Ds #arrowhead path{fill:#333;stroke:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .sequenceNumber{fill:#fff}#mermaid-svg-aeBu7OBWqCKz38Ds #sequencenumber{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds #crosshead path{fill:#333;stroke:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .messageText{fill:#333;stroke:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .labelBox{stroke:#ccf;fill:#ECECFF}#mermaid-svg-aeBu7OBWqCKz38Ds .labelText,#mermaid-svg-aeBu7OBWqCKz38Ds .labelText>tspan{fill:#000;stroke:none}#mermaid-svg-aeBu7OBWqCKz38Ds .loopText,#mermaid-svg-aeBu7OBWqCKz38Ds .loopText>tspan{fill:#000;stroke:none}#mermaid-svg-aeBu7OBWqCKz38Ds .loopLine{stroke-width:2px;stroke-dasharray:2, 2;stroke:#ccf;fill:#ccf}#mermaid-svg-aeBu7OBWqCKz38Ds .note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-aeBu7OBWqCKz38Ds .noteText,#mermaid-svg-aeBu7OBWqCKz38Ds .noteText>tspan{fill:#000;stroke:none}#mermaid-svg-aeBu7OBWqCKz38Ds .activation0{fill:#f4f4f4;stroke:#666}#mermaid-svg-aeBu7OBWqCKz38Ds .activation1{fill:#f4f4f4;stroke:#666}#mermaid-svg-aeBu7OBWqCKz38Ds .activation2{fill:#f4f4f4;stroke:#666}#mermaid-svg-aeBu7OBWqCKz38Ds .mermaid-main-font{font-family:"trebuchet ms", verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .section{stroke:none;opacity:0.2}#mermaid-svg-aeBu7OBWqCKz38Ds .section0{fill:rgba(102,102,255,0.49)}#mermaid-svg-aeBu7OBWqCKz38Ds .section2{fill:#fff400}#mermaid-svg-aeBu7OBWqCKz38Ds .section1,#mermaid-svg-aeBu7OBWqCKz38Ds .section3{fill:#fff;opacity:0.2}#mermaid-svg-aeBu7OBWqCKz38Ds .sectionTitle0{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .sectionTitle1{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .sectionTitle2{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .sectionTitle3{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .sectionTitle{text-anchor:start;font-size:11px;text-height:14px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .grid .tick{stroke:#d3d3d3;opacity:0.8;shape-rendering:crispEdges}#mermaid-svg-aeBu7OBWqCKz38Ds .grid .tick text{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .grid path{stroke-width:0}#mermaid-svg-aeBu7OBWqCKz38Ds .today{fill:none;stroke:red;stroke-width:2px}#mermaid-svg-aeBu7OBWqCKz38Ds .task{stroke-width:2}#mermaid-svg-aeBu7OBWqCKz38Ds .taskText{text-anchor:middle;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .taskText:not([font-size]){font-size:11px}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutsideRight{fill:#000;text-anchor:start;font-size:11px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutsideLeft{fill:#000;text-anchor:end;font-size:11px}#mermaid-svg-aeBu7OBWqCKz38Ds .task.clickable{cursor:pointer}#mermaid-svg-aeBu7OBWqCKz38Ds .taskText.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutsideLeft.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutsideRight.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-aeBu7OBWqCKz38Ds .taskText0,#mermaid-svg-aeBu7OBWqCKz38Ds .taskText1,#mermaid-svg-aeBu7OBWqCKz38Ds .taskText2,#mermaid-svg-aeBu7OBWqCKz38Ds .taskText3{fill:#fff}#mermaid-svg-aeBu7OBWqCKz38Ds .task0,#mermaid-svg-aeBu7OBWqCKz38Ds .task1,#mermaid-svg-aeBu7OBWqCKz38Ds .task2,#mermaid-svg-aeBu7OBWqCKz38Ds .task3{fill:#8a90dd;stroke:#534fbc}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutside0,#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutside2{fill:#000}#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutside1,#mermaid-svg-aeBu7OBWqCKz38Ds .taskTextOutside3{fill:#000}#mermaid-svg-aeBu7OBWqCKz38Ds .active0,#mermaid-svg-aeBu7OBWqCKz38Ds .active1,#mermaid-svg-aeBu7OBWqCKz38Ds .active2,#mermaid-svg-aeBu7OBWqCKz38Ds .active3{fill:#bfc7ff;stroke:#534fbc}#mermaid-svg-aeBu7OBWqCKz38Ds .activeText0,#mermaid-svg-aeBu7OBWqCKz38Ds .activeText1,#mermaid-svg-aeBu7OBWqCKz38Ds .activeText2,#mermaid-svg-aeBu7OBWqCKz38Ds .activeText3{fill:#000 !important}#mermaid-svg-aeBu7OBWqCKz38Ds .done0,#mermaid-svg-aeBu7OBWqCKz38Ds .done1,#mermaid-svg-aeBu7OBWqCKz38Ds .done2,#mermaid-svg-aeBu7OBWqCKz38Ds .done3{stroke:grey;fill:#d3d3d3;stroke-width:2}#mermaid-svg-aeBu7OBWqCKz38Ds .doneText0,#mermaid-svg-aeBu7OBWqCKz38Ds .doneText1,#mermaid-svg-aeBu7OBWqCKz38Ds .doneText2,#mermaid-svg-aeBu7OBWqCKz38Ds .doneText3{fill:#000 !important}#mermaid-svg-aeBu7OBWqCKz38Ds .crit0,#mermaid-svg-aeBu7OBWqCKz38Ds .crit1,#mermaid-svg-aeBu7OBWqCKz38Ds .crit2,#mermaid-svg-aeBu7OBWqCKz38Ds .crit3{stroke:#f88;fill:red;stroke-width:2}#mermaid-svg-aeBu7OBWqCKz38Ds .activeCrit0,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCrit1,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCrit2,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCrit3{stroke:#f88;fill:#bfc7ff;stroke-width:2}#mermaid-svg-aeBu7OBWqCKz38Ds .doneCrit0,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCrit1,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCrit2,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCrit3{stroke:#f88;fill:#d3d3d3;stroke-width:2;cursor:pointer;shape-rendering:crispEdges}#mermaid-svg-aeBu7OBWqCKz38Ds .milestone{transform:rotate(45deg) scale(0.8, 0.8)}#mermaid-svg-aeBu7OBWqCKz38Ds .milestoneText{font-style:italic}#mermaid-svg-aeBu7OBWqCKz38Ds .doneCritText0,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCritText1,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCritText2,#mermaid-svg-aeBu7OBWqCKz38Ds .doneCritText3{fill:#000 !important}#mermaid-svg-aeBu7OBWqCKz38Ds .activeCritText0,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCritText1,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCritText2,#mermaid-svg-aeBu7OBWqCKz38Ds .activeCritText3{fill:#000 !important}#mermaid-svg-aeBu7OBWqCKz38Ds .titleText{text-anchor:middle;font-size:18px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds g.classGroup text{fill:#9370db;stroke:none;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:10px}#mermaid-svg-aeBu7OBWqCKz38Ds g.classGroup text .title{font-weight:bolder}#mermaid-svg-aeBu7OBWqCKz38Ds g.clickable{cursor:pointer}#mermaid-svg-aeBu7OBWqCKz38Ds g.classGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-aeBu7OBWqCKz38Ds g.classGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds .classLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.5}#mermaid-svg-aeBu7OBWqCKz38Ds .classLabel .label{fill:#9370db;font-size:10px}#mermaid-svg-aeBu7OBWqCKz38Ds .relation{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-aeBu7OBWqCKz38Ds .dashed-line{stroke-dasharray:3}#mermaid-svg-aeBu7OBWqCKz38Ds #compositionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #compositionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #aggregationStart{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #aggregationEnd{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #dependencyStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #dependencyEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #extensionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds #extensionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds .commit-id,#mermaid-svg-aeBu7OBWqCKz38Ds .commit-msg,#mermaid-svg-aeBu7OBWqCKz38Ds .branch-label{fill:lightgrey;color:lightgrey;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .pieTitleText{text-anchor:middle;font-size:25px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .slice{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds g.stateGroup text{fill:#9370db;stroke:none;font-size:10px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds g.stateGroup text{fill:#9370db;fill:#333;stroke:none;font-size:10px}#mermaid-svg-aeBu7OBWqCKz38Ds g.statediagram-cluster .cluster-label text{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds g.stateGroup .state-title{font-weight:bolder;fill:#000}#mermaid-svg-aeBu7OBWqCKz38Ds g.stateGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-aeBu7OBWqCKz38Ds g.stateGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-aeBu7OBWqCKz38Ds .transition{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-aeBu7OBWqCKz38Ds .stateGroup .composit{fill:white;border-bottom:1px}#mermaid-svg-aeBu7OBWqCKz38Ds .stateGroup .alt-composit{fill:#e0e0e0;border-bottom:1px}#mermaid-svg-aeBu7OBWqCKz38Ds .state-note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-aeBu7OBWqCKz38Ds .state-note text{fill:black;stroke:none;font-size:10px}#mermaid-svg-aeBu7OBWqCKz38Ds .stateLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-aeBu7OBWqCKz38Ds .edgeLabel text{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-aeBu7OBWqCKz38Ds .node circle.state-start{fill:black;stroke:black}#mermaid-svg-aeBu7OBWqCKz38Ds .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-aeBu7OBWqCKz38Ds #statediagram-barbEnd{fill:#9370db}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-state .divider{stroke:#9370db}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-aeBu7OBWqCKz38Ds .note-edge{stroke-dasharray:5}#mermaid-svg-aeBu7OBWqCKz38Ds .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-aeBu7OBWqCKz38Ds .error-icon{fill:#522}#mermaid-svg-aeBu7OBWqCKz38Ds .error-text{fill:#522;stroke:#522}#mermaid-svg-aeBu7OBWqCKz38Ds .edge-thickness-normal{stroke-width:2px}#mermaid-svg-aeBu7OBWqCKz38Ds .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-aeBu7OBWqCKz38Ds .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-aeBu7OBWqCKz38Ds .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-aeBu7OBWqCKz38Ds .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-aeBu7OBWqCKz38Ds .marker{fill:#333}#mermaid-svg-aeBu7OBWqCKz38Ds .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-aeBu7OBWqCKz38Ds {color: rgba(0, 0, 0, 0.75);font: ;}推出推出推出幾乎處處收斂依概率收斂均方收斂依分布收斂

參考資料

[1] 劉嘉焜，王公恕，應(yīng)用隨機(jī)過(guò)程（第二版）。

總結(jié)

以上是生活随笔為你收集整理的依分布收敛、依概率收敛、均方收敛、几乎处处收敛的全部?jī)?nèi)容，希望文章能夠幫你解決所遇到的問(wèn)題。

如果覺(jué)得生活随笔網(wǎng)站內(nèi)容還不錯(cuò)，歡迎將生活随笔推薦給好友。

概率