## 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**:

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

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

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.