binomial distribution

Binomial properties:

1. A series of trails, classed as being either ‘success‘ or ‘failure‘ (Bernoulli trial).
2. The trials are identical and independent. ( in the sense that the outcome of one trial has no effect on the outcome of any other)
3. The random variable $X$ denotes the number of successes obtained in the n trials.

Sample space for an experiment in which $n = 3$:

$$S = { fff, sff, fsf, ffs, ssf, sfs, sss , fss }$$

Note that:

$$begin{align} P[X = 0] &= (1-p)^3\ P[X = 1] &= 3 cdot p(1-p)^2\ P[X = 2] &= 3 cdot p^2(1-p)\ P[X = 3] &= p^3\ end{align}$$

Definition ( Binomial distribution):

$$f(x) = left( begin{array} { l } { n } \ { x } end{array} right) p^x(1-p)^{n-x} qquad x=0,1,2,...,n qquad 0<p<1$$

We write: $X sim operatorname{B}(n, p)$

Specially, $operatorname{B}(1, p )= operatorname{Bernoulli}(p)$

Binomial moment generating function:

$$m_X(t) = (q+pe^t)^n qquad q = 1-p$$

Expectation and Variance for the Binomial Distribution:

$$E[X] = mu = np qquad and qquad operatorname{Var} [X] = sigma^2 = npq$$