Lecture 4. C) Binomial
Binomial Distribution
A r.v. [math]X[/math] follows a Binomial distribution with parameters [math]n\in\mathbb{N}[/math], [math]p\in\left[0,1\right][/math]if [math]X[/math] is discrete with pmf
[math]f_{X}\left(\left.x\right|n,p\right)=\begin{cases} \left(\begin{array}{c} n\\ x \end{array}\right)p^{x}\left(1-p\right)^{n-x}, & if\,x\in\left\{ 0,1,...,n\right\} \\ 0, & otherwise \end{cases}[/math]
where [math]\left(\begin{array}{c} n\\ x \end{array}\right)[/math] is called the binomial coefficient, and is defined by [math]\left(\begin{array}{c} n\\ x \end{array}\right)=\frac{n!}{x!\left(n-x\right)!}[/math]
The Binomial Distribution characterizes the number of successes out of [math]n[/math] binary (Bernoulli) experiments. Parameter [math]n[/math] is the number of trials, [math]p[/math] is the probability of success, and [math]x[/math] is the realized number of successes. If [math]X\sim Bin\left(1,p\right)[/math], then [math]X\sim Ber\left(p\right)[/math].
Mean
[math]E\left(X\right)=np[/math]
Variance
[math]Var\left(X\right)=np\left(1-p\right)[/math]
MGF
[math]M_{X}\left(t\right)=\left(1-p+p\exp\left(t\right)\right)^{n}[/math]
Notice that the expressions above are clearly related to their Bernoulli analogues.
