Binomial Distribution
A discrete random variable $X$ is said to have Binomial distribution with parameter $n$ and $p$ if its probability mass function is
$$ \begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \binom{n}{x} p^x q^{n-x}, & \hbox{$x=0,1,2,\cdots, n$;} \\ & \hbox{$0<p<1, q=1-p$} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
Mean of Binomial Distribution
The mean or expected value of binomial random variable $X$ is
$E(X) = np$.
Variance of Binomial distribution
The variance of Binomial random variable $X$ is
$V(X) = npq$.
For Binomial distribution Mean > Variance.
Moment Generating Function of Binomial Distribution
The moment generating function (MGF) of Binomial distribution is given by
$M_X(t) = (q+pe^t)^n.$
Cumulant Generating Function of Binomial Distribution
The cumulant generating function of Binomial random variable $X$ is
$K_X(t) = n\log_e (q+pe^t)$.
Probability Generating Function of Binomial Distribution
The probability generating function (PGF) of Binomial distribution is given by