均匀分布

Uniform Distribution $X\sim U(a,b)$

\dfrac{1}{b-a} \quad a<x<b\\ \;\;0\quad \quad \quad 其他 \end{cases}$$ $$F(x)=\begin{cases} \quad 0\quad \quad \quad \,x<a\\ \dfrac{x-a}{b-a}\quad a\leq x<b\\ \quad 1\quad \quad \quad \,x\geq b \end{cases}$$ <svg xmlns="http://www.w3.org/2000/svg" version="1.1" viewBox="0 0 698.2107444454932 222.6980271056109" width="698.2107444454932" height="222.6980271056109"> <!-- svg-source:excalidraw --> <defs> <style class="style-fonts"> @font-face { font-family: "Virgil"; src: url("https://excalidraw.com/Virgil.woff2"); } @font-face { font-family: "Cascadia"; src: url("https://excalidraw.com/Cascadia.woff2"); } @font-face { font-family: "Assistant"; src: url("https://excalidraw.com/Assistant-Regular.woff2"); } </style> </defs> <rect x="0" y="0" width="698.2107444454932" height="222.6980271056109" fill="#ffffff"></rect><g stroke-linecap="round"><g transform="translate(11.1558195754792 167.45685708326982) rotate(0 149.58841371639357 0.014718154306592623)"><path d="M0.2 1 C50.05 1.19, 249.54 0.16, 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16.09999999999991)"><text x="0" y="25.760546874999996" font-family="Helvetica, Segoe UI Emoji" font-size="28px" fill="#1e1e1e" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">F(x)</text></g><g stroke-linecap="round"><g transform="translate(402.16394715637944 171.16588568226052) rotate(0 98.3117238190265 -55.166191552697)"><path d="M0.47 -0.15 C33.36 -18.6, 164.82 -91.81, 197.64 -110.1 M-0.74 -1.28 C32.05 -20.16, 164.26 -93.65, 197.22 -112.1" stroke="#1e1e1e" stroke-width="1" fill="none"></path></g></g><mask></mask><g stroke-linecap="round"><g transform="translate(599.5853748005507 59.45507232518503) rotate(0 43.858440954547746 0.488536396150721)"><path d="M-0.03 -0.86 C14.48 -0.4, 72.04 1.56, 86.53 2.04 M-1.5 1.3 C13.4 1.51, 73.48 0.65, 88.63 0.47" stroke="#1e1e1e" stroke-width="1" fill="none"></path></g></g><mask></mask><g stroke-linecap="round"><g transform="translate(464.46987194933604 59.759764009316314) rotate(0 67.34253863877223 -0.5508638528115171)"><path d="M0.84 -0.19 C23.25 -0.2, 111.06 -0.65, 133.5 -0.91" stroke="#e03131" stroke-width="1.5" fill="none" stroke-dasharray="8 9"></path></g></g><mask></mask><g transform="translate(443.002765927959 48.5835393051716) rotate(0 5.071291390060765 10.486308588122029)"><text x="0" y="16.778449933542483" font-family="Helvetica, Segoe UI Emoji" font-size="18.237058414125364px" fill="#e03131" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">1</text></g><g transform="translate(137.52059521763022 59.10491216406945) rotate(0 31.616679084150746 11.37543577436486)"><text x="0" y="18.20108363286186" font-family="Helvetica, Segoe UI Emoji" font-size="19.78336656411267px" fill="#1e1e1e" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">1/(a+b)</text></g></svg> **期望：** $E(X)=\dfrac{a+b}{2}$ **方差：** $D(X)=\dfrac{(b-a)^{2}}{12}$ 均匀分布表示取值**等可能**或者完全随机的随机变量 通过均匀分布的随机数可以产生其他分布的随机数，是 [[蒙特卡洛方法\|蒙特卡洛方法]]模拟的基础 二维均匀分布 $(X,Y)\sim U(D)$ $$f(x,y)=\begin{cases} \dfrac{1}{A} \quad (x,y)\in D\\ \\ 0\quad \quad (x,y) \not\in D \end{cases}$$ $$\begin{align} P\left\{(X,Y) \in D_{1} \right\}=\iint \limits_{D_{1}} \dfrac{1}{A} d\sigma =\dfrac{S_{D_{1}}}{A} \end{align}$$