Bernoulli distribution
From Freepedia
| Probability mass function | |
| Cumulative distribution function | |
| Parameters | <math>p>0\,</math> (real) <math>q\equiv 1-p\,</math> |
| Support | <math>k=\{0,1\}\,</math> |
| pmf | <math>
\begin{matrix}
q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1
\end{matrix}
</math>
|
| cdf | <math>
\begin{matrix}
0 & \mbox{for }k<0 \\q & \mbox{for }0<k<1\\1 & \mbox{for }k>1
\end{matrix}
</math>
|
| Mean | <math>p\,</math> |
| Median | N/A |
| Mode | <math>\textrm{max}(p,q)\,</math> |
| Variance | <math>pq\,</math> |
| Skewness | <math>\frac{q-p}{\sqrt{pq |
| Kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| mgf | {{{mgf}}} |
| Char. func. | {{{char}}} |
kurtosis =<math>\frac{3p^2+p+1}{pq}\,</math>|
entropy =<math>-q\ln(q)-p\ln(p)\,</math>|
mgf =<math>q+pe^t\,</math>|
char =<math>q+pe^{it}\,</math>|
}}
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability <math>p</math> and value 0 with failure probability <math>q=1-p</math>. So if X is a random variable with this distribution, we have:
- <math> \Pr(X=1) = 1- \Pr(X=0) = p\!</math>.
The probability mass function f of this distribution is
- <math> f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\
1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.</math>
The expected value of a Bernoulli random variable X is <math>EX=p</math>, and its variance is
- <math>\textrm{var}(X)=p(1-p)\,</math>.
The Bernoulli distribution is a member of the exponential family.
Related distributions
- If <math>X_1,\cdots,X_n</math> are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then <math>Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)</math> (binomial distribution).
