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## Proposition: Binomial Distribution

We repeat a Bernoulli experiment $$n$$ times. Each time, we observe, if an event $$A$$ occurred or not. Let the probability of $$A$$ to occur be constantly $$p:=p(A)$$ during each repetition of the experiment. Let $$X$$ be the random variable counting the number $$k$$ of the realizations of $$A$$. Clearly, $$A$$ can occur for $$k$$ times with $$0\le k\le n$$. The probability mass function of $$A$$ occurring exactly $$k$$ times is given by

$p(X = k)=\begin{cases} \binom nk p^k(1-p)^{n-k}&\text{for }k=0,1,\ldots n\\\\ 0&\text{else.}\end{cases}$

The binomial distribution (i.e. the probability distribution of the random variable $$X$$) is given by

$$\begin{array}{rcll} p(X \le x)&=&0&\text{for }x < 0\\ p(X \le x)&=&\sum_{k=0}^{k=x}\binom nk p^k(1-p)^{n-k}&\text{for }0\le x < n\\ p(X \le x)&=&1&\text{for }x \ge n\\ \end{array}$$

In the following interactive you can change the probability $$p$$ of observing an event $$A$$ in a Bernoulli experiment repeated $$20$$ times. Changing this probability will change the probability mass function for different values of $$X$$ (red) of observing $$A$$ from $$0$$ to $$20$$ times and the probability distribution (blue):

| | | | | created: 2014-02-23 22:45:41 | modified: 2018-05-23 22:05:34 | by: bookofproofs | references: [856], [1796]