Binomial Distribution

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.

\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.