binomial_distribution
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binomial_distribution [2019/11/04 15:01] – created hkimscil | binomial_distribution [2020/11/27 19:37] – hkimscil | ||
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====== Binomial Distribution ====== | ====== Binomial Distribution ====== | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | {n \choose x} = \displaystyle \frac {n!}{x!(n-x)!} | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | **The number of successes in n independent Bernoulli trials has a binomial distribution.** | ||
+ | |||
+ | n independent Bernoulli trials | ||
+ | * There are n independent trials | ||
+ | * Each trial can result in one of two possible outcomes, labelled success and failure. | ||
+ | * success can be a bad thing -- tire blow-up. | ||
+ | * P(success) = p, | ||
+ | * P(failure) = 1-p | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | P(X=x) = _{n}C_{x} \cdot p^{x} \cdot (1-p)^{n-x}, | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | A balanced dice is rolled 3 times. What is probability a 5 comes up exactly twice? | ||
+ | |||
+ | p = 1/6 | ||
+ | n = 3 | ||
+ | x = 2 | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | P(X=2) & = & {{3} \choose {2}} \left(\frac{1}{6}\right)^{2} \left(\frac{5}{6}\right)^{3-2} \\ | ||
+ | & = & 0.0694 | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | < | ||
+ | > dbinom(2, 3, 1/6) | ||
+ | [1] 0.06944444 | ||
+ | > | ||
+ | </ | ||
+ | |||
+ | |||
+ | |||
\begin{eqnarray*} | \begin{eqnarray*} | ||
X \sim B(n, p) \\ | X \sim B(n, p) \\ | ||
\end{eqnarray*} | \end{eqnarray*} |
binomial_distribution.txt · Last modified: 2020/11/27 19:42 by hkimscil