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--- binomial_distribution [2019/11/04 15:01] – created hkimscil
+++ binomial_distribution [2019/11/04 15:28] – hkimscil
@@ Line 1: / Line 1: @@
 ====== 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*}
+<code>
+> dbinom(2, 3, 1/6)
+[1] 0.06944444
+>
+</code>
 \begin{eqnarray*}
 X \sim B(n, p) \\
 \end{eqnarray*}