Normal Approximation in R-Code

Abstract

The aim of this research is to understand when a normal distribution can be approximated along with a discrete distribution. Sometimes it may be easier to approximate the binomial distribution as well. At the same time, it’s important to remember that while the normal distribution is continuous, the binomial distribution is discrete. This study is also aimed at estimating how normal distribution can be used to approximate the Poisson distribution. We also attempted to consider binomial probabilities with the use of the binomial coefficient, in order to create a simple formula. Given that the formula includes factorials, it may cause certain difficulties, so the normal approximation can help with the computation.

Introduction

Parameters of the binomial distribution (n and p) determine its shape. The parameter p represents the probability of a failure or success. If it’s small, or in case q = 1- p is small, the distribution will be asymmetrical. In case p is almost equal 0.5, and n is bigger enough, we can use the normal distribution to approximate the binomial distribution. The binomial distribution is symmetrical, and its shape looks like a bell. If the p parameter is too small or close to 1, we still can use a normal distribution, in case n is very big. Moreover, if n is big enough, we can use both the normal approximation and the Poisson approximation. However, the choice must be made in favor of the normal approximation, because cumulative probabilities are calculated easily with the use of tables and a number of other supported technologies. In case samples are too big, it may cause many difficulties in calculating certain probabilities. In order to approximate the probabilities of success, we should use the normal distribution. Otherwise, such a process may require much more complex computations. If n is big enough, for example, if it exceeds 20, and p isn’t close to 1 or 0, the normal distribution can be applied.

R programming helps calculate probabilities for normal, binomial, and Poisson distributions. R code allows us not only to test the input, but also to model the output graphically. The use of R programming requires an operating system that is able to perform calculations of any kind.

- First of all, we have to consider the conditions that allow the discrete distributions to be close to normal distributions.

- The next step is to make the discreet distribution incline towards a bell shape by generating a set. R can be used to specify the necessary size.

- The generated distribution must be compared with the normal distribution.

Continuity adjustment is corrected to approximate a discrete distribution. A random variable takes any real values within an interval. At the same time, a discrete random variable can be equal only to a number of specified values, so in case we use the normal distribution for the approximation of the binomial, the probabilities are approximated more precisely.

Poisson Approximation of Binomial Probabilities

If n is big, but p is small, the binomial distribution can be approximated with the help of Poisson distribution. The Poisson approximation allows us to obtain results which are very close to the exact probability. We also can use the normal approximation to obtain the same probability. Given that the normal distribution is used for a continuous random variable, and the binomial distribution is applied for a discrete random variable, we need a continuity correction to approximate a discrete distribution with a normal distribution.

The Normal Approximation to the Poisson Distribution

The Poisson distribution can be approximated by the normal distribution, but only in case the parameter λ is big enough. If so, for example, if λ is bigger than 15, we can use the normal distribution in approximation: X~N(λ, λ). However, such an approach also requires a continuity correction, because a discrete distribution is approximated with a continuous distribution.

Conclusion

This study proved that the use of the normal distribution in the approximation of the binomial distribution allows us to obtain more accurate results. Moreover, the binomial distribution inclines towards the normal distribution, with an increase in the value of n. We can also conclude that the approximation of binomial distribution with the normal one is a particular case of a wider phenomenon. We also proved that the continuity adjustment must be corrected. According to the results of the research, R programming helps obtain more accurate outcomes of distributions.

The use of the normal distribution in approximation can be quite a useful practice, in case we create necessary conditions. Otherwise, such an approximation may not be the best solution for the estimation of probabilities.