| Version 2 |
Version 1 |
|
Consider an experiment with two possible outcomes (success and failure), which happen randomly. Let $p$ be the probability of success. If the experiment is repeated $n$ times, the probability of having exactly $x$ successes is
|
Consider a random experiment with two possible outcomes (success and failure), which happen randomly. Let $p$ be the probability of success. If the experiment is repeated $n$ times, the probability of having exactly $x$ successes is
|
|
$$f(x)=\left({n\atop x}\right)p^x(1-p)^{(n-x)}.$$
|
$$f(x)=\left({n\atop x}\right)p^x(1-p)^(n-x).$$
|
| The distribution function determined by the probability function $f(x)$ is called a \emph{Bernoulli distribution} or \emph{binomial distribution}. |
The distribution function determined by the probability function $f(x)$ is called a \emph{Bernoulli distribution} or \emph{binomial distribution}. |
| Here are some plots for $f(x)$ with $n=20$ and $p=0.3$, $p=0.5$. |
Here are some plots for $f(x)$ with $n=20$ and $p=0.3$, $p=0.5$. |
| \figuraex{binom10p3}{scale=0.75} |
\figuraex{binom10p3}{scale=0.75} |
| \figuraex{binom10p5}{scale=0.75} |
\figuraex{binom10p5}{scale=0.75} |
| The corresponding distribution function is |
|
| $$F(n)=\sum_{k\leq x}{n\atop k}p^k q^{n-k}$$ |
|
| where $q=1-p$. Notice that if we calculate $F(n)$ we get the binomial expansion for $(p+q)^n$, this being the reason for the distribution being called binomial. |
|
