Binomial Distribution: Definition, Density function, properties and application

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Binomial distribution is a special case of Bernoulli distribution where the number of trial is up to n times instead of two times ( probability of success “p” and probability of failure “q”).


binomial distribution

Binomial distribution was discovered by James Bernoulli (1654-1705) in the year 1700 qnd was first published posthumously in 1713, eight years after his death. Let a random experiment be performed repeatedly, each repitition being called a trial and let the occurrence of an event in a trial be called a success and its non-occurrence a failure. Consider a set of n (finite)  independent Bernoulli trials in which the probability ‘p’ of success in any trial is constant for each trial, then q=1-p, is the probability of failure in any trial.

 Let X be a discrete random variable, X is said to have binomial distribution if the density of X is defined  as,

    \[ f(X=r;n,p)=\left ( _{r}^{n}\textrm{} \right )p^{r}q^{1-r}; where, (_{r}^{n}\textrm{})=\frac{n!}{r!(n-r)!} for.. r=0,1,2,...,n \]


Here , n = total number of observation

           r = number of trial

           p = probability of success

           q = probability of failure

 So the probability distribution of X is called the binomial distribution. This is a discrete probability distribution.


The two independent constant n and p in the distribution are known as the parameter of the distribution. ‘n’ is also sometimes, known as the degree of the binomial distribution.
Binomial distribution is a discrete distribution as X can take only the integral value, 0,1,2,…,n. Any random variable which follow binomial distribution is known as binomial variate.
binomial curve


  • Fixed number of trials, n, which means that the experiment is repeated a specific number of times.
  • The n trials are independent, which means that what happens on one trial does not influence the outcomes of other trials.
  • There are only two outcomes, which are called a success and a failure.
  • The probability of a success doesn’t change from trial to trial, where p = probability of success and q = probability of failure, q = 1  p .

Mean and Variance of Binomial distribution

 The mean of the binomial distribution is the expectation of X which is as following,

                     Mean(X)= µ = E(X) = np

The variance of the binomial distribution is ,

                  Variance(X)=σ2 = V (X) = np(1 p) .

 Application of binomial distribution

In practical life we use binomial distribution when want to know the occurrence of an event. For example-

  • Manufacturing company uses binomial distribution to detect the defective goods or items.
  • In clinical trail binomial trial is used to detect the effectiveness of the drug.
  • Moreover binomial trail is used in various field such as market research.

Shape of Binomial Distribution

The shape or pattern of binomial distribution depends on the values of p and n. If p=q=0.5, the distribution will be symmetrical regardless of the values of n. If p≠q, the distribution will be asymmetrical. Given a particular n, the more the difference between p and q , the greater the skewness of the distribution will be. When p<q , the distribution will be positively skewed and when p>q then the distribution will be negatively skewed. However as the value of n increases, the distribution will become less and less skewed. When n becomes infinitely large, the distribution will approach symmetry irrespective of the difference between p and q.
The effects of increases in n on the shape of the binomial distribution are shown in the below figure. The first set of figures is drawn  on the assumption that p=q. The distribution is always symmetrical. As the value of n increases, the bars become narrower and more numerous. As n approaches infinity, the bars become vertical lines with no space in between, and the distribution becomes bell-shaped smooth curve.
And the second set of figures in drawn on the assumption that the probability of a success on a single trial is 0.1. It shows that as n become larger and larger, the skewness of the distribution disappears and in the long run the distribution becomes continuous. Thus it is apparent that as the value of n increases, the binomial distribution becomes a continuous and symmetrical distribution whether or not p and q are equal.

 Real Life examples of Binomial Distribution

There are many, many excellent examples —

  • We can see from baby that born, there are two options between boy or girl and binomial distribution was used to predict that baby is girl or boy.
  • I thought of involves the traffic light at the intersection of my street with the main road. I pass by there every morning. Sometimes the light is green, and I make it through; sometimes it’s red, and I don’t. The number of times I make the light each month is probably a decent binomial distribution.
  • In a manufacturing context, the number of faulty items in a batch of products might follow a binomial distribution, if the probability of failures is constant. In practice, this probably isn’t true, because of the chance that an equipment failure generates clusters of failures. I vaguely remember doing a project about 45 years ago on failures of welds, where the number of failures in a particular product was in practice very close to the expected binomial distribution, with something like n=40 and p=0.01 (roughly 67% 0 fails, 27% 1 fail, 6% more than 1).
  • A chart of height for adult males in any given country, or adult females in any given country, will follow a binomial distribution (the famous “bell curve”) beautifully, in which the curve is very high around the average and then tapers off in either direction. Exactly as the binomial distribution predicts, there will be a few “outliers”… people exceptionally above or below average, but their numbers will be extremely small, approaching zero as you get farther away from the average.
  • A chart of IQ for people of any given age will also tend to have a binomial distribution… again, showing the famous “bell curve,” which bulges around the middle. Although charting is a different phenomenon, it will tend to have the same mathematical properties as height charts.

Here’s a binomial distribution you can produce yourself… but you can do it a thousand times faster by writing a computer program that generates random numbers.

Flip a coin 100 times per session. Record the number of heads. This number should be close to 50 per session, but it won’t always be 50 exactly. Typically, most sessions will produce results fairly close to 50… let’s say from 40 to 60.

Now, greater divergence from 50 is possible (such as 65 or 35) and will in fact occur, given enough repeated sessions. But the farther you get away from 50, the rarer such results become.

So if you do many of these sessions and chart the results, you will again see the famous “bell-curve”… bulging around the average value (in this case, 50) and then tapering off as you get farther and farther away from the average.

This shape of curve is exactly what the Binomial Theorem predicts and you can see that by looking at the numbers for a row of Pascal’s Triangle, especially rows that are at least five or six down from the top.




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