\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amsmath, amssymb}
\usepackage{soul}
\usepackage[dvipsnames]{xcolor}
\newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}}
\begin{document}
\begin{align*}
\binom n i p^i ( 1 - p )^{ n - i }
& = \frac{ n! }{ i! ( n - i )! } \, p^i \, ( 1 - p )^{ n - i } \\
& = \frac{ n! }{ \mathcolorbox{ProcessBlue}{ i! } ( n - i )! }
\left ( \frac \lambda n \right )^i
\left ( 1 - \frac \lambda n \right )^{ n - i } \\
& = \mathcolorbox{YellowGreen}{ \frac{ n! }{ n^i ( n - i )! } }
\frac{ \lambda^i }{ \mathcolorbox{ProcessBlue}{ i! } }
\frac{ \mathcolorbox{Yellow}{ \left ( 1 - \frac \lambda n \right )^n }
\to e^{ -\lambda } \text{ when } n \to +\infty }{
\left \{ \left ( 1 - \dfrac \lambda n \right )^i \right \}
\to 1 \text{ when } n \to +\infty } \\
& = \mathcolorbox{Yellow}{ e^{ -\lambda } }
\frac{ \lambda^i }{ i! }
\mathcolorbox{YellowGreen}{
\frac{ n ( n - 1 ) \cdots ( n - i + 1 ) }{ n^i } } \to 1
\text{ when } n \to +\infty \\
& = \frac{ \lambda^i }{ i! } e^{ -\lambda }
\end{align*}
\end{document}
-编辑-
首先,颜色并没有什么特别的含义。只有红框内最右边的部分才有意义。
如你所见,分数线也跨越了\to e^{ -\lambda } \text{ when } n \to +\infty
和\to 1 \text{ when } n \to +\infty
注释;相反,我只希望它覆盖\left ( 1 - \frac \lambda n \right )^n
和\left ( 1 - \frac \lambda n \right )^i
答案1
这是一个带有\overbrace
和\underbrace
指令的解决方案。
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{mathtools, amssymb}
\begin{document}
\begin{align*}
\binom{n}{i} p^i ( 1-p )^{ n-i }
&= \frac{ n! }{ i! ( n-i )! } \, p^i \, ( 1-p )^{ n-i }\\
&= \frac{ n! }{ i ( n-i )! }
\left( \frac{\lambda}{n}\right)^{\!i}
\left( 1-\frac{\lambda}{n}\right)^{\!n-i} \\[\jot]
&= \frac{ n! }{ n^i ( n-i )! } \,
\frac{ \lambda^i }{ i! } \,
\frac{ \overbrace{\left( 1 - \frac{\lambda}{n} \right)^{\!\!n}}%
^{\mathclap{\to e^{-\lambda} \text{ as } n\to\infty}}
}{
\underbrace{\left( 1 - \frac{\lambda}{n} \right)^{\!\!i}}%
_{\mathclap{\to 1 \text{ as } n\to\infty}}
} \\[\jot]
&= e^{ -\lambda } \,
\frac{ \lambda^i }{ i! } \,
{\underbrace{\frac{ n (n-1) \dotsb(n-i+1) }{ n^i }}%
_{\to 1\text{ as } n \to \infty}} \\
&= \frac{ \lambda^i }{ i! }\, e^{-\lambda }
\end{align*}
\end{document}