# Binomial Distribution

Posted by Beetle B. on Thu 18 May 2017

## Bernoulli Distribution

A Bernoulli random variable is one whose only possible values are 0 and 1.

$$P(X=1)=p$$

$$E[X]=p,V[X]=p(1-p)$$

## Binomial Distribution

Requirements for a binomial experiment:

1. There are a fixed number of trials: $$n$$
2. The trials are identical, and their outcome can be classified into two states (traditionally “success” and “failure”).
3. The trials are all independent.
4. The probability of success is constant: $$p$$

A binomial random variable is where we count the number of passes in a binomial experiment.

\begin{equation*} b(x;n,p)=\binom{n}{x}p^{x}(1-p)^{n-x} \end{equation*}

for $$x=0,1,\dots,n$$.

Note that it will sum to 1 using the expansion of $$(a+b)^{n}$$.

The cdf is denoted by $$B(x;n,p)$$

The distribution is symmetric when $$p=0.5$$

### Mean, Variance

$$E(X)=np$$, $$V(X)=np(1-p)$$

Note that the variance is maximum when $$p=0.5$$

### Sampling Without Replacement

Note: If you sample without replacement, you do not have a binomial experiment! But if $$n<<N$$, then $$p$$ is roughly the same from trial to trial and you can treat it as binomial.

As a rule of thumb: You can treat it as one when the sample size is less than 5% of the population.