Normal and Binomial Distribution

Introduction

In a given set of events, there is always a certain probability of something happening. If a set of data is given that describes certain events then certain events have the possibility of occurring due to the occurrence of the first set of events. Normal and binomial distribution helps in finding the probability of two events taking place.

What are the Normal Distribution and Binomial Distribution?

Continuous data that cause a symmetric distribution is defined as a normal distribution. It has the characteristics of a ‘bell’ shape. From a finite sample, the distillation of data in its binary form is known as Binomial distribution. The calculation of binomial distribution provides the possibility of achieving r events from the n trial. Similarly, another distribution is named Poisson distribution. This distribution defines the distribution of binary data from an infinite sample. Therefore, it provides the possibility of achieving r events inside a population.

More about Normal Distribution

With the help of a couple of parameters, the normal distribution can be described. The two parameters are μ and σ. μ represents the mean of the population or the central point of distribution, and σ represents the standard deviation of the population. The distribution is done symmetrically around the mean. Small values of populations that have the standard deviation σ possess a concentrated form of distribution near the center μ. However, those that contain large standard deviations have their distribution scattered all over the measurement axis. One of the mathematical properties of the normal distribution is that 95% of all the distributions are located between Μ−(1.96xσ) and μ+(1.96xσ). By changing the multiplier from 1.96 to 2.58 a specified percentage of 99% of all the normal distributions that are located in the intervals which are corresponding.  While calculating, both the parameters of normal distribution, μ, and σ, should be determined by observing the sample data. To serve this purpose a sample of the population is taken randomly. Then the calculation of the sample standard deviation and the sample mean takes place. When such sample is derived from a normal distribution and given that the sample is not very small, then around 95% of the sample within the interval can be calculated by

X¯−[1.96×SD(x¯)] to X¯+[1.96×SD(x¯)]

This calculation is performed by only substituting the parameters of the population with the estimates of the sample as per the previous expression. In favorable situations, this interval can be used for estimation of the reference interval of a specific laboratory test which can be used for diagnostic.

More about binomial distribution

Suppose there is a group of crops that are given a fertilizer that has been newly introduced. The fertilizer is supposed to be a growth boost for plants. In such a case, the portion that is provided with the fertilizer successively can be signified by the alphabet p. This portion can be treated as the estimation of the success rate of the fertilizer. Therefore, the p is the sample portion that is analogous to the sample mean. So, if the score of s crops that are not successful in growing, is zero and the score of the portion that was successful is 1, then p equals r/n. Here n signifies the total number of crops that are treated and the p has symbolized as the mean. The binomial distribution formula states that

P(x:n,p) = ⁿC _x P_x(1-p)^n-x

Where,

n = denotes the number of experiments.

x = 0, 1, 2, 3, 4, …

p = The probability of achieving success in one experiment,

p = Probability of facing a failure in one experiment (= 1 – p).

Conclusion

The normal and binomial distribution is important to calculate the mean and also find the possibilities of any given event. Therefore, these distribution systems can act as a tool to find answers to the probability of uncertain events. Due to its importance, in competitive examinations like GATE, such questions are asked. Students aspiring to ace the GATE examinations must have a proper understanding of the binomial distribution and normal distribution. The binomial distribution is important to know because it is applied in real life as well. For instance, when a new medicine is introduced in the market, it has two probabilities. Either the medicine is going to cure a disease or it is not going to cure a disease. Similarly, suppose an individual buys a lottery ticket. There are two possibilities. Either he will win or he will lose. Binomial distribution helps in finding the probabilities of these situations.