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<title>指数分布期望自测 | 银行等待时间例题</title>
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.card{background:#fff;border-radius:16px;padding:24px;margin-bottom:20px;box-shadow:0 4px 16px #00000010;}
h2{color:#2563eb;margin-top:0;font-size:1.3rem;}
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.tip{color:#64748b;font-size:0.9rem;}
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<div class="card">
  <h2>📖 原题：顾客银行等待时间（指数分布）</h2>
  <div class="question">
设顾客在某银行的窗口等待服务的时间 $X$（以分计）服从指数分布，其概率密度：
$$
f(x)=
\begin{cases}
\displaystyle \frac15 e^{-x/5}, & x>0\\[4pt]
0, & x\le 0
\end{cases}
$$
试求顾客等待服务的平均时间（期望 $E(X)$）？
  </div>
  <div class="input-row">
    <span>你的答案：</span>
    <input id="ansInput" type="number" step="0.01" placeholder="输入期望数值">
    <button onclick="checkAnswer()">提交校验</button>
    <button class="btn-secondary" onclick="toggleAnswer()">查看答案与推导</button>
  </div>
  <div id="checkResult"></div>
  <div id="answerBlock" class="answer-block">
    <strong>✅ 标准答案：$\boldsymbol{E(X)=5}$（分钟）</strong>
    <br><br>
    指数分布通用形式：$f(x)=\lambda e^{-\lambda x},\;x>0$，期望 $E(X)=\dfrac1\lambda$
    <br>
    本题对比：$\lambda=\dfrac15$，所以 $E(X)=\dfrac{1}{1/5}=5$
    <br><br>
    积分严格推导：
    $$
    \begin{aligned}
    E(X)&=\int_{0}^{+\infty} x\cdot f(x)dx
    =\int_{0}^{+\infty} x\cdot \frac15 e^{-x/5} dx \\
    &=-x e^{-x/5}\bigg|_{0}^{+\infty}+\int_{0}^{+\infty}e^{-x/5}dx \\
    &=0+\Big[-5e^{-x/5}\Big]_{0}^{+\infty}=5
    \end{aligned}
    $$
  </div>
</div>

<div class="card">
  <h2>📊 交互式图像调节面板（指数分布 $f(x)=\lambda e^{-\lambda x}$）</h2>
  <div class="slider-row">
    <label>速率参数 $\lambda =$ <span id="lamVal">0.20</span> &nbsp;（原题 λ=0.2）</label>
    <input type="range" id="lamSlider" min="0.05" max="2.0" step="0.01" value="0.2" oninput="updateChart()">
    <div class="tip">拖动滑块修改 λ，期望 $E(X)=1/\lambda$ 自动更新，图中红色竖线 = 期望位置</div>
  </div>
  <div class="chart-wrap">
    <canvas id="expChart"></canvas>
  </div>
</div>

<script>
// 标准答案
const trueAns = 5;
let chart;
let lam = 0.2;

function toggleAnswer(){
  const block = document.getElementById("answerBlock");
  block.style.display = block.style.display === "block" ? "none" : "block";
}

function checkAnswer(){
  const val = parseFloat(document.getElementById("ansInput").value);
  const resDom = document.getElementById("checkResult");
  if(isNaN(val)){
    resDom.innerHTML = "<span class='result-wrong'>请输入合法数字！</span>";
    return;
  }
  if(Math.abs(val - trueAns) < 0.001){
    resDom.innerHTML = "<span class='result-correct'>✅ 正确！期望就是5分钟</span>";
  }else{
    resDom.innerHTML = `<span class='result-wrong'>❌ 不对，正确期望是 ${trueAns}</span>`;
  }
}

function updateChart(){
  lam = parseFloat(document.getElementById("lamSlider").value);
  document.getElementById("lamVal").innerText = lam.toFixed(2);
  const mu = 1 / lam; //期望
  //生成x数据
  const xs = [];
  const pdfYs = [];
  const cdfYs = [];
  const maxX = Math.max(mu * 4,10);
  for(let x=0;x<=maxX;x+=0.15){
    xs.push(x);
    const pdf = lam * Math.exp(-lam * x);
    pdfYs.push(pdf);
    const cdf = 1 - Math.exp(-lam * x);
    cdfYs.push(cdf);
  }
  if(chart) chart.destroy();
  const ctx = document.getElementById("expChart").getContext("2d");
  chart = new Chart(ctx,{
    type:"line",
    data:{
      labels:xs,
      datasets:[
        {label:"概率密度 PDF $f(x)$",data:pdfYs,borderColor:"#2563eb",fill:false,pointRadius:0,borderWidth:2},
        {label:"分布函数 CDF $F(x)$",data:cdfYs,borderColor:"#10b981",fill:false,pointRadius:0,borderWidth:2}
      ]
    },
    options:{
      responsive:true,maintainAspectRatio:false,
      plugins:{
        title:{display:true,text:`指数分布 λ=${lam.toFixed(2)}，期望 E(X)=${mu.toFixed(3)}`,font:{size:14}},
        annotation:{
          annotations:{
            expectLine:{type:"line",xMin:mu,xMax:mu,borderColor:"#ef4444",borderWidth:2,label:{display:true,content:"E(X)期望",position:"top"}}
          }
        }
      },
      scales:{x:{title:{display:true,text:"x 等待时间"}},y:{title:{display:true,text:"f(x) / F(x)"}}}
    }
  })
}

//初始化
updateChart();
</script>
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