In this post, we will learn about Binomial Distribution and its basics.

A statistical experiment having successive independent trials having two possible outcomes (such as success and failure; true and false; yes and no; right and wrong etc.) and probability of success is equal for each trial, while this kind of experiment is repeated a fixed number of times (say $n$ times) is called ** Binomial Experiment**, Each

*trial*of this

*Binomial experiment*is known as

*Bernoulli trial*(a trial which is a single performance of an experiment), for example.

### Properties of the Binomial Experiment

- Each
*trial*of*the Binomial Experiment*can be classified as*a success*or*failure*. - The
*probability of success*for each trial of the experiment is equal. - Successive
*trials are independent*, that is, the occurrence of one outcome in an experiment does not affect the occurrence of the other. - The experiment is
*repeated**a**fixed number of times*.

**Binomial Distribution**

Let $X$ be a *discrete random variable*, which denotes the *number of successes* of a *Binomial Experiment* (we call this binomial random variable). The *random variable* assumes isolated values as $X=0,1,2,\cdots,n$. The probability distribution of the *binomial random variables is termed **binomial distribution*. It is a *discrete probability distribution*.

**Binomial Probability Mass Function**

The probability function of the binomial distribution is also called binomial probability mass function and can be denoted by $b(x, n, p)$, that is, a binomial distribution of random variable $X$ with $n$ (given number of trials) and $p$ (probability of success) as parameters. If $p$ is the probability of success (alternatively $q=1-p$ is probability of failure such that $p+q=1$) then probability of exactly $x$ success can be found from the following formula,

\begin{align}

b(x, n, p) &= P(X=x)\\

&=\binom{n}{x} p^x q^{n-x}, \quad x=0,1,2, \cdots, n

\end{align}

where $p$ is the probability of success of a single trial, $q$ is the probability of failure and $n$ is the number of independent trials.

The formula gives the probability for each possible combination of $n$ and $p$ of a binomial random variable $X$. Note that it does not give $P(X <0)$ and $P(X>n)$. The binomial distribution is suitable when $n$ is small and is applied when sampling done is with replacement.

\[b(x, n, p) = \binom{n}{x} p^x q^{n-x}, \quad x=0,1,2,\cdots,n,\]

is called Binomial distribution because its successive terms are exactly the same as that of binomial expansion of

\begin{align}

(q+p)^n=\binom{0}{0} p^0 q^{n-0}+\binom{n}{1} p^1 q^{n-1}+\cdots+\binom{n}{n-1} p^n q^{n-(n-1)}+\binom{n}{n} p^n q^{n-n}

\end{align}

$\binom{n}{0}, \binom{n}{1}, \binom{n}{2},\cdots, \binom{n}{n-1}, \binom{n}{n}$ are called Binomial coefficients.

Note that it is necessary to describe the limit of the random variable otherwise, it will be only the mathematical equation, not the probability distribution.

Read more about Binomial Probability Distribution and Take Online MCQ tests on Probability Distributions