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Mathematical Models |
A mathematical model is a mathematical idealization in the form of a system, proposition, formula, or equation of a physical, biological, or social phenomenon. Thus, a theoretical, perfectly balanced die that can be tossed in a purely random fashion is a mathematical model for an actual physical die. The probability that in n throws of a mathematical die a throw of 6 will occur k times is
in which (¥) is the symbol for the binomial coefficient
The statistician confronted with a real physical die will devise an experiment, such as tossing the die n times repeatedly, for a total of Nn tosses, and then determine from the observed throws the likelihood that the die is balanced and that it was thrown in a random way.
In a related but more involved example of a mathematical model, many sets of measurements have been found to have the same type of frequency distribution. For example, let x1, x2, …, xN be the number of 6's cast in the N respective runs of n tosses of a die and assume N to be moderately large. Let y1, y2, …, yN be the weights, correct to the nearest 1/100 g, of N lima beans chosen haphazardly from a 100-kg bag of lima beans. Let z1, z2, …, zN be the barometric pressures recorded to the nearest 1/1000 cm by N students in succession, reading the same barometer. It will be observed that the x's, y's, and z's have amazingly similar frequency patterns. The statistician adopts a model that is a mathematical prototype or idealization of all these patterns or distributions. One form of the mathematical model is an equation for the frequency distribution, in which N is assumed to be infinite:
in which e (approximately 2.7) is the base for natural logarithms (see Logarithm). The graph of this equation (Fig. 4) is the bell-shaped curve called the normal, or Gaussian, probability curve. If a variate x is normally distributed, the probability that its value lies between a and b is given by
The mean of the x's is 0, and the standard deviation is 1. In practice, if N is large, the error is exceedingly small.
