Computational Aspects of Normal Distribution

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•H.P.Gautam
The purpose of this article is not to explain any more the usefulness of normal distribution in decision-making process no matter whether in social sciences or in natural sciences. Nor is the purpose of making any discussions on the theory of how it can be derived. The only objective of writing this article is to acquaint the enthusiastic readers (specially students) with the simple procedure ( iterative procedure) for finding the numerical value of a normally distributed variable. The procedure is simple in the sense that the students even from non-mathematical background can easily use the technique discussed below to find the value, and it will not an exaggeration to say that he or she after going through the article not only can compute an individual value but also can generate the whole table for such values. At most he require a scientific calculator.

It should be bone in mind that a person using normal distribution as an analytical tool need not be familiar with the computational aspect of normal distribution as various computer package and statistical tables are available in the market. However, he must know how the distribution helps him take the right decisions.

Though one need not compute the numerical values himself rather than to how to use and interpret these values, yet he will have a deeper understanding how normal distribution works, if he becomes familiar with the computational process too. In this context, the whole discussion is divided into 3 sections. The first section begins with analysis of function of the distribution, while the second one deals with the general procedure for computational technique. Lastly, in the third section a practical example will be solved to justify what we have told in the first and second section of the article. In addition, a computer program will be listed which gives the result up to the correct 4 decimal places

Section-1 Normal distribution function
The equation of normal curve is defined as
(eqn. 1)
From the above equation we can derive the following density function between two finite values of x (x1 and x2). (eqn. 2)
From the purpose of simplification , transform x into z and find the integration as given below for the standard variate . Under this transformation of x to normal variate z the above equation changes to . The above integration can be expanded as follows. We know that . Replacing the value in above eqn. 2 , the new modified eqn.2 that is F(z) looks like series of integrations as shown below eqn.-3

Most of normal tables show the area (probability) between z1=0 and z2=m (any arbitrary value positive or negative). We also compute the probability between z1=0 and z2=m. For the purpose, we integrate the above terms separately and then after substitute the upper limit of z (z2=m) and lower limits of z (z1=0). After substitution, the eqn. 3 can be rewritten as eqn.4

where k is the number of terms.
If one considers the following properties of above series of f(m) or eqn. 4, he finds that the above series is convergent. The properties of the series are
-the value of
-the value of

-the value of kth term = .
The common multiplier value is . Let us analyze the trend of multiplier value. The value of in the multiplier remains constant throughout the series as the value of m is arbitrary value of z which is fixed for a given series. The value of changes from one term to another term of series. The value of denominator is greater than that of numerator implying that decreases as the number of terms(k) increases as long as k>1. This proves that absolute value of multiplier decreases monotonically and the series is convergent for the finite number of terms. For example the multiplier for 2nd term in order to get 3rd term is equal to and the multiplier for 3rd term to get 4th term is equal to . It is clearly seen that the absolute value of is less than that of...
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