# 伯努利分布

参数 ${\displaystyle 1>p>0\,}$（实数） ${\displaystyle k=\{0,1\}\,}$ ${\displaystyle {\begin{matrix}q&{\mbox{for }}k=0\\p~~&{\mbox{for }}k=1\end{matrix}}}$ ${\displaystyle {\begin{matrix}0&{\mbox{for }}k<0\\q&{\mbox{for }}0\leq k<1\\1&{\mbox{for }}k\geq 1\end{matrix}}}$ ${\displaystyle p\,}$ N/A ${\displaystyle {\begin{matrix}0&{\mbox{if }}q>p\\0,1&{\mbox{if }}q=p\\1&{\mbox{if }}q ${\displaystyle pq\,}$ ${\displaystyle {\frac {q-p}{\sqrt {pq}}}}$ ${\displaystyle {\frac {6p^{2}-6p+1}{p(1-p)}}}$ ${\displaystyle -q\ln(q)-p\ln(p)\,}$ ${\displaystyle q+pe^{t}\,}$ ${\displaystyle q+pe^{it}\,}$

• 概率質量函數為：
• ${\displaystyle f_{X}(x)=p^{x}(1-p)^{1-x}=\left\{{\begin{matrix}p&{\mbox{if }}x=1,\\q\ &{\mbox{if }}x=0.\\\end{matrix}}\right.}$
• 期望值為：
• ${\displaystyle \operatorname {E} [X]=\sum _{i=0}^{1}x_{i}f_{X}(x)=0+p=p}$
• 方差為：
• ${\displaystyle \operatorname {Var} [X]=\sum _{i=0}^{1}(x_{i}-\operatorname {E} [X])^{2}f_{X}(x)=(0-p)^{2}(1-p)+(1-p)^{2}p=p(1-p)=pq}$

## 参考文献

1. ^ Sheldon M Ross. 《Introduction to probability and statistics for engineers and scientists》. Academic Press. 2009: 第141頁. ISBN 9780123704832.