Binomial Distribution

Topics: Probability - Bernoulli Distribution

---

(definition)

Let $X$ be a random variable that handles the repetition of Bernoulli trial ($X$: the number of successful trials). Let $n$ be the number of independent Bernoulli trials and $p$ be the probability of success.

In such a case, we say that $X$ has a binomial distribution and write it as $X\sim \mathrm{Bin}(n,p)$.

Single vs. Multiple Trials

If $X$ handles multiple Bernoulli trials (of the same nature), then it has a binomial distribution.

On the other hand, when it handles a single Bernoulli trial, it has a Bernoulli distribution.

Density Function §

(theorem)

The density function of an $X$ such that $X\sim \mathrm{Bin}(n,p)$ distribution is given by:

$$f_X(x)=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}& \text{if }x\in \left\{0,\dots ,n\right\}\\ 0& \text{otherwise}\end{array}$$

Expected Value §

(theorem)

The expected value of such an $X$ is:

$$\mathbb{E}[X]=np$$

Variance §

(theorem)

The variance of such an $X$ is:

$$\mathrm{Var}(X)=np(1-p)$$

Moment-Generating Function §

(theorem)

Finally, the moment-generating function of such an $X$ is:

$$M_X(t)=[(1-p)-p{e}^t{]}^n$$