## Binomial Probability Distribution

To learn about Binomial Probability Distribution, we first need to understand the Bernoulli Trials.

Bernoulli Trials

Many experiments consist of repeated independent trials and each trial has only two possible outcomes such as head or tail, right or wrong, alive or dead, defective or non-defective, etc. If the probability of each outcome remains the same (constant) throughout the trials, then such trials are called the Bernoulli Trials.

### Binomial Probability Distribution

Binomial Probability Distribution is a discrete probability distribution describing the results of an experiment known as the Bernoulli Process. The experiment having n Bernoulli trials is called a Binomial Probability experiment possessing the following four conditions/ assumptions

1. The experiment consists of $n$ repeated tasks.
2. Each trial results in an outcome that may be classified as success or failure.
3. The probability of success denoted by $p$ remains constant from trial to trial.
4. The repeated trials are independent.

A Binomial trial can result in a success with probability p and a failure with probability $1-p$ having $n-x$ number of failures, then the probability distribution of Binomial Random Variable, the number of successes in $n$ independent trial is:

\begin{align*}
P(X=x)&=\binom{n}{x} \, p^x \, q^{n-x} \\
&=\frac{n!}{x!(n-x)!}\, p^x \, q^{n-x}
\end{align*}

The Binomial probability distribution is the most widely used distribution in situations of two outcomes. It was discovered by the Swiss mathematician Jakob Bernoulli (1654—1704) whose main work on “the ars Conjectandi” (the art of conjecturing) was published posthumously in Basel in 1713.

Mean of Binomial Distribution:   Mean = $\mu = np$

Variance of Binomial Distribution:  Variance = $npq$

Standard Deviation of Binomial Distribution:  Standard Deviation = $\sqrt{npq}$

Moment Coefficient of Skewness:

\begin{align*}
\beta_1 &= \frac{q-p}{\sqrt{npq}}  \\
&= \frac{1-2p}{\sqrt{npq}}
\end{align*}

Moment Coefficient of Kurtosis:  $\beta_3 = 3+\frac{1-6pq}{npq}$

Generate Binomial Random Numbers in R Language