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<h3>Functional relationships for <script type="math/tex">\;X \sim Binomial(n,p),\;\;x=0,1,2,...,n;\;\;0\leq\;p\leq\;1</script>
</h3>
$$
\begin{aligned}
f(x|n,p)&=\dbinom{n}{x}p^{x}(1-p)^{n-x}\\\\\\\\
F(x|n,p)&=\sum_{i=0}^{x}\dbinom{n}{i}(p)^{i}(1-p)^{n-i}\\\\\\\\
E[X]&=np\\\\\\\\
Var[X]&=np(1-p)
\end{aligned}
$$
