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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}, \;\; \text{for} \;\; x = 0, 1, 2, . . ., n. \\ \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}} (\frac{1}{6})^{2} (\frac{5}{6})^{3-2} \\ & = & 0.0694 \end{eqnarray*}

> dbinom(2, 3, 1/6)
[1] 0.06944444
> 

\begin{eqnarray*} X \sim B(n, p) \\ \end{eqnarray*}

binomial_distribution.txt · Last modified: 2019/11/04 15:28 by hkimscil