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--- binomial_distribution [2019/11/04 15:01] – created hkimscil
+++ binomial_distribution [2020/11/27 19:42] (current) – hkimscil
@@ Line 1: / Line 1: @@
 ====== Binomial Distribution ======
+  - 1번의 시행에서 특정 사건 A가 발생할 확률을 p라고 하면
+  - n번의 (독립적인) 시행에서 사건 A가 발생할 때의 확률 분포를
+  - 이항확률분포라고 한다.
+\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.**
+이는 n 번의 독립적인 Bernoulli trials 로 볼 수 있다.
+  * 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
+일반적으로 binomial distribution은 아래와 같이 계산된다.
+\begin{align*}
+P(X=x) & = _{n}C_{x} \cdot p^{x} \cdot (1-p)^{n-x}, \;\; \text{for} \;\; x = 0, 1, 2, . . ., n. \\
+\text{or } & \\
+P(X=x) & = {{n} \choose {x}} \cdot p^{x} \cdot (1-p)^{n-x}, \;\; \text{for} \;\; x = 0, 1, 2, . . ., n. \\
+\end{align*}
+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*}
+<code>
+> dbinom(2, 3, 1/6)
+[1] 0.06944444
+>
+</code>
 \begin{eqnarray*}
 X \sim B(n, p) \\
 \end{eqnarray*}