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 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.