The most important thing to remember is that all sequences of numbers can be classified into those which are equal to each other. Such a relationship is called a binomial relationship. It is important to note, however, that even in such cases, there are no exact matches. Therefore, the most useful technique for studying the relationships between sequences is to examine the number distribution and find the common distribution.
The binomial distribution is a normal distribution whose power spectrum is proportional to the square root of the number of cases. If we plot out the binomial distribution for any number n, we will get a plot of the frequency distributions, which are usually shaped like a bell curve. In a binomial distribution, the number of times it appears in a set of sample numbers is related to the normal distribution function, n(r), of the distribution of all the numbers, are, at the same time. The number of times that each number occurs, however, depends on the binomial distribution function.
To learn about the binomial distribution, we need to understand its relationship to the normal distribution. When the normal distribution function is plotted against the binomial distribution function, they form a triangle, with a smaller and flatter part of the triangle denoting the normal distribution function, and the larger and higher sides of the triangle denoting the binomial distribution function. By this relationship, if there is an excess of n’s in the sample, the ratio of the frequencies n(r) of the number in the set of samples is greater than the normal distribution function.
The size of the triangle is dependent upon the number of cases, because the sum of the squares of the triangles of all the cases is less than the square of the triangle of the sets of samples. This is true for most cases. In some cases, however, there are cases where the number of cases exceeds the number of samples, so that the sum of the squares is less than the square of the samples. That is, the normal distribution function cannot fit neatly around the binomial distribution function.
The next step is to find out what kinds of numbers are present in the data set. After finding out the normal distribution function, we may use the binomial distribution function to find out the frequency distribution of the numbers. That is, we can determine whether or not the numbers are evenly distributed or unevenly distributed, with one group of numbers having more cases than another.
A probability distribution is a kind of distribution, which is used to describe the distribution of frequencies as a function of the number of instances of that number. A probability distribution has a shape like a normal curve and is called a power spectrum, which plots the values of frequencies against the number of instances of that number.
Another type of distribution function is the binomial or Gaussian distribution function. There is also a beta distribution, which plots the data from a set against the number of samples and gives a curve that extends from the lowest values to the highest values. The beta distribution is a better fit than the binomial distribution in many cases, but it tends to have too many tails, so that the curves are not so clearly curved.