DETERMINATION OF SOME STATISTICAL CHARACTERISTICS FOR DETERMINING THE STOCHASTIC RELATIONSHIP BETWEEN TWO RANDOM VARIABLES ON THE “MINSK-1” COMPUTER

G. L. KALINICHENKO

(Presented by the scientific-technical conference of the AVTF)

Problem Statement

Suppose that, instead of a two-dimensional theoretical distribution law, a sample of \(n\) observations with two possible values is given. Each of the observations records a pair of associated values of the quantities being studied. All these \(n\) observations are entered into a correlation table, i.e., into a two-entry table (Table 1). Each row consists of possible values for a discrete random variable or of \(m\)-intervals for a continuous random variable \(X\), for a definite value of the random variable \(Y\). Similarly, each column consists of \(k\)-intervals for the random variable \(Y\) at a definite value of the random variable \(X\). In this table \(mk\) cells are filled in. In the cell corresponding to the \(j\)-th row and the \(i\)-th column, the number of observations with possible values \(X=x_i\) and \(Y=y_j\) is entered; we shall denote it by \(n_{ij}\). If the random variable is continuous, then in this cell is entered the number of observations for which

\[ x_i-\frac{\Delta x_i}{2} < x < x_i+\frac{\Delta x_i}{2} \quad \text{and} \quad y_j-\frac{\Delta y}{2} < y < y_j+\frac{\Delta y}{2}. \]

The task of the present work is to compile a standard program for the “Minsk-1” computer, with the largest possible working field and without the use of external storage, which would make it possible to give the researcher an answer concerning the existence and degree of stochastic relationship between two random variables under study.

Content of the work:

A standard program has been compiled for the “Minsk-1” computer which prints a table of values for the empirical regression line of \(y\) on \(x\) and \(X\) on \(y\). Empirical initial moments of the random variables \(X\) and \(Y\) up to and including the 4th order are found. This makes it possible to print arithmetic means, root-mean-square values of the random variables \(X\) and \(Y\), as well as coefficients of skewness and kurtosis, which allow one to obtain a first approximate idea of the closeness of the distribution under study to the normal distribution. The principal values, however, are the correlation coefficient and the coefficients of the correlation ratios, which make it possible to judge the presence and degree of stochastic relationship between two random variables. The standard program was compiled using the interpreting system of the “Minsk-1” computer with floating-point numbers recorded in one cell, used in the computing laboratory of the Tomsk Polytechnic Institute [2].

The greater part of the machine memory, from cell 0710 to cell 1777, is allocated for

initial data and makes it possible to process any rectangular matrix that can be placed in 564 cells.

The number of observations, however, is practically unlimited. Cells 0001 through 0436 are occupied by the interpreting program for writing floating-point numbers into one cell; cells 0440 through 0471 by the standard square-root extraction program; and cells 0472 through 0473 by the program itself, with working cells and pseudocodes for program self-restoration.

Before each input of new information, cells 0656 through 0661 are filled, as are cells 0502 and 0503.

where the number of columns is \(m\), and \(n\) is the number of rows of the matrix in the octal number system.

Into cells 0701 through 0707, the following corresponding numbers are entered in the decimal system:

\[ y_0,\ \Delta y,\ x_0,\ \Delta x_0,\ 2,\ 3,\ 4. \]

Program start from cell 0472.

This program has been debugged and is used in the computing laboratory of the Tomsk Polytechnic Institute.

Program for the “Minsk-1” Computer

REFERENCES

1. I. V. Dunin-Barkovskii and N. V. Smirnov. Probability Theory and Mathematical Statistics. “Nauka,” Moscow, 1969.

2. V. D. Del’, G. L. Kalinichenko, V. A. Mal’tsev. Automation of programming for the “Minsk-1” digital computer by the method of interpreting systems. Tomsk, 1969.