伯努利分布参数 |
(实数) |
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值域 |
![{\displaystyle k=\{0,1\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/531e2ca3ed846ec08cd2c9623e8a81f34243ffb4) |
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概率质量函数 |
![{\displaystyle {\begin{matrix}q&{\mbox{for }}k=0\\p~~&{\mbox{for }}k=1\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58bfe4980dbb194c5c897dca9d72dc872ec2e0f7) |
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累積分布函數 |
![{\displaystyle {\begin{matrix}0&{\mbox{for }}k<0\\q&{\mbox{for }}0\leq k<1\\1&{\mbox{for }}k\geq 1\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb75441ffdb76fc62fbc94723c0b86c8f8241e8) |
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期望值 |
![{\displaystyle p\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa5f88a712eb9b03398066a0577fdcf33e02c6) |
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中位數 |
N/A |
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眾數 |
![{\displaystyle {\begin{matrix}0&{\mbox{if }}q>p\\0,1&{\mbox{if }}q=p\\1&{\mbox{if }}q<p\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c07eaa4bfc0837baf39cd1e4914347cd7f0d495d) |
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方差 |
![{\displaystyle pq\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1669a5ebf9af3cf97f4b52512cd9af1eceee587e) |
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偏度 |
![{\displaystyle {\frac {q-p}{\sqrt {pq}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08fca3adeb465e259b3ee4ab4292f7058ee2c8b4) |
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峰度 |
![{\displaystyle {\frac {6p^{2}-6p+1}{p(1-p)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/189a0d20b85eba635b94330030f7d2490d4b1300) |
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熵 |
![{\displaystyle -q\ln(q)-p\ln(p)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea75df2fc9fe0c5d5b747477efcf7fee3380a151) |
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矩生成函数 |
![{\displaystyle q+pe^{t}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28262bb4c07dd358133e0eca6c0dc56151c6998e) |
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特徵函数 |
![{\displaystyle q+pe^{it}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210c63881c775461dc60b1df161983a8e8102baf) |
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伯努利分布的图像表示
伯努利分布(英語:Bernoulli distribution),又名两点分布或者0-1分布,是一個離散型概率分布,為紀念瑞士科學家雅各布·伯努利而命名。若伯努利試驗成功,則伯努利隨机變量取值為1。若伯努利試驗失敗,則伯努利隨机變量取值為0。記其成功概率為
,失敗概率為
。[1]則
- 其概率質量函數為:
![{\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33224c832a3cfdc7d5bb93bb37d2d3994b97f496)
- 其期望值為:
![{\displaystyle \operatorname {E} [X]=\sum _{i=0}^{1}x_{i}f_{X}(x)=0+p=p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe1f0f8ce4525c70a88bab81eb7d6bbc4dff77d)
- 其方差為:
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1c7410b1881d54d37b73f1d35d700245dd7a21)
- ^ Sheldon M Ross. 《Introduction to probability and statistics for engineers and scientists》. Academic Press. 2009: 第141頁. ISBN 9780123704832.