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2.1.7 The Binomial distribution function

Assume a single trial that can take only two outcomes (``Yes'' and ``No'' or 0 and 1 or ``Head'' and ``Tail'' etc.). Such a variable is called a Binary or Bernoulli random variable. Now, assume that such a single trial is repeated $ n$ times. A Binomial (discrete) variable $ X$ is the number of ``successes'' in these $ n$ trials and its' probability distribution describes the Probability of each number $ \{0,1,2,3,\ldots \}$.

Definition 11   The Binomial distribution function with parameters $ n$ (# trials) and $ p$ (probability of ``success'') is:

$\displaystyle P_X(k) = ( n \choose k ) p^k (1-p)^{n-k} $


$\displaystyle (n \choose k ) = \frac{n!}{(n-k)!k!} = \frac{n (n-1) (n-2) (n-3) \dots 2\dot 1}{(n-k)(n-k-1)\dots 1 k(k-1)(k-2)\dots 1} $

Which is the number of ways to choose a subset of size $ k$ from a set of size $ n$.

Mean and Variance

These graphshows $ 3$ examples (different $ n$ and $ p$ parameters) of the binomial distributions (a graph the distribution function with a few parameter.

Eran Borenstein