(prerequisite-Poisson distribution) Binomial Distribution

• Binomial Distribution

 

It is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. 

 

$$\begin{array}{}& Pr(K=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-<!--\; -\; -->p{)}^{n-<!--\; -\; -->k}\end{array}$$

- $k=0,1,2,\cdots <!--\; \cdots \; -->,n$

$$\begin{array}{}& (\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-<!--\; -\; -->k)!}=n{\mathrm{C}}_k\end{array}$$

- $p$ : Success Probability

- $n$ : The number of events occurring within a given interval of time or space.

- $\left(p^k\right)$ : The success of the $k$ times.

- $((1-<!--\; -\; -->p{)}^{n-<!--\; -\; -->k})$ : The probability of $n-<!--\; -\; -->k$ times failure.

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Example.

 

$k=0,1,2,\cdots <!--\; \cdots \; -->,10$ : The number of times to toss the coin.

$p$ : The probability of a HEAD coming out when you toss a coin, assume 1/2.

$$\begin{array}{}& k=0:\frac{10!}{0!\cdot <!--\; \cdot \; -->10!}{\left(\frac{1}{2}\right)}^0\cdot <!--\; \cdot \; -->{(1-<!--\; -\; -->\frac{1}{2})}^{10}=0.0010\end{array}$$ $$\begin{array}{}& k=1:\frac{10!}{1!\cdot <!--\; \cdot \; -->9!}{\left(\frac{1}{2}\right)}^1\cdot <!--\; \cdot \; -->{(1-<!--\; -\; -->\frac{1}{2})}^9=0.0098\end{array}$$ $$\begin{array}{}& k=2:\frac{10!}{2!\cdot <!--\; \cdot \; -->8!}{\left(\frac{1}{2}\right)}^2\cdot <!--\; \cdot \; -->{(1-<!--\; -\; -->\frac{1}{2})}^8=0.0439\end{array}$$ $$⋮<!--\; ⋮\; -->$$ $$\begin{array}{}& k=10:\frac{10!}{10!\cdot <!--\; \cdot \; -->0!}{\left(\frac{1}{2}\right)}^{10}\cdot <!--\; \cdot \; -->{(1-<!--\; -\; -->\frac{1}{2})}^0=0.0010\end{array}$$

$$\begin{array}{}& Pr(K=k)=[0.0010,0.0098,0.0439,0.1172,0.2051,0.2461,0.2051,0.1172,0.0439,0.0098,0.0010]\end{array}$$

 

https://angeloyeo.github.io/2021/04/23/binomial_distribution.html