Normal distribution and its parameters
Random variables are associated with random events. Random events are said to occur when it is impossible to predict unambiguously the result that can be obtained under certain conditions.
Suppose we throw an ordinary coin. Usually the result of this procedure is not unambiguously determined. It can only be stated with certainty that one of two things will happen: either the "eagle" or "tails" will drop out or "tails". Any of these events will be random. You can enter a variable that describes the outcome of this random event. Obviously, this variable will take two discrete values: & quot; eagle & quot; and & quot; Tare & quot ;. Since we can not foresee in advance exactly which of the two possible values this variable will take, we can say that in this case we are dealing with random variables.
Suppose now that in the experiment we make an estimate of the reaction time of the subject under presentation of some stimulus. As a rule, it turns out that even when the experimenter will take all measures to ensure that the standardization of experimental conditions, minimizing or even reducing to zero the possible variation in the presentation of the stimulus, the measured values of the test reaction time will still vary. In this case, it is said that the reaction time of the subject is described by a random variable. Because, in principle, in the experiment, we can get any time of the reactions - the set of possible values of the reaction time, which can be obtained as a result of the measurement, is infinite, - say continuous of the random variable .
The question arises: are there any regularities in the behavior of random variables? The answer to this question is affirmative.
So, if you hold an infinitely large number of tosses of the same coin, you can find out that the number of depositions of each of the two sides of the coin will be approximately the same, unless, of course, the coin is false and not bent. To emphasize this pattern, introduce the concept of the probability of a random event. It is clear that in the case of a coin toss, one of the two possible events will happen without fail. This is due to the fact that the total probability of these two events, otherwise called total probability, is 100%. Assuming that both of the two events associated with the coin test occur with equal shares of probability, then the probability of each outcome separately appears to be 50%. Thus, theoretical reflections allow us to describe the behavior of a given random variable. Such a description in mathematical statistics is denoted by the term distribution of a random variable .
The situation is more complicated with a random variable that does not have a clearly defined set of values, i.e. turns out to be continuous. But in this case, we can also note some important patterns of its behavior. Thus, by conducting an experiment with the measurement of the reaction time of the subject, it can be noted that the various intervals of the reaction time of the subject are evaluated with varying degrees of probability. Most likely, it is rare when a subject reacts too quickly. For example, in problems of a semantic solution, the subjects practically can not respond more or less accurately with a speed of less than 500 ms (1/2 s). Similarly, it is unlikely that the subject, conscientiously following the instructions of the experimenter, will greatly delay his response. In problems of the semantic solution, for example, reactions estimated more than 5 s are usually regarded as unreliable. Nevertheless, with 100% certainty it can be assumed that the reaction time of the subject will be in the range from 0 to + oo. But this probability is made up of the probabilities of each individual value of the random variable. Therefore, the distribution of a continuous random variable can be described as a continuous function y = f ( x ).
If we are dealing with a discrete random variable, when all its possible values are known in advance, as in the coin example, it is usually not very difficult to build a model of its distribution. It is enough to introduce only some reasonable assumptions, as we did in this example. The situation with the distribution of continuous magnitudes assuming a previously unknown number of values is more complicated. Of course, if we, for example, developed a theoretical model describing the behavior of the subject in an experiment with the measurement of the reaction time in solving the semantic solution problem, one could try on the basis of this model to describe the theoretical distribution of specific values of the reaction time of the same subject when one and the same stimulus. However, this is not always possible. Therefore, the experimenter may be forced to assume that the distribution of the random variable of interest is described by some law already studied in advance. Most often, although this may not always turn out to be absolutely correct, the so-called normal distribution is used for these purposes, acting as a standard for the distribution of any random variable, regardless of its nature. This distribution was first described mathematically in the first half of the XVIII century. de Moivrom.
Normal distribution occurs when the phenomenon of interest to us is influenced by an infinite number of random factors balancing each other. Formally, the normal distribution, as shown by de Moivre, can be described by the following relationship:
where x is the random variable of interest to us, the behavior of which we investigate; P is the probability value associated with this random variable; π and e - known mathematical constants describing the ratio of the circumference to the diameter and the base of the natural logarithm; μ and σ2 are the parameters of the normal distribution of the random variable, respectively, the mathematical expectation and variance of the random variable x.
To describe the normal distribution, it is necessary and sufficient to determine only the parameters μ and σ2.
Therefore, if we have a random variable whose behavior is described by equation (1.1) with arbitrary values of μ and σ2, we can designate it as Ν ( μ, σ2) without storing all the details of this equation.
Fig. 1.1. Single normal distribution (z-distribution)
Any distribution can be visualized as a graph. The graphically normal distribution has the form of a bell-shaped curve, the exact shape of which is determined by the distribution parameters, i.e. mathematical expectation and variance. The parameters of the normal distribution can take practically any value that is limited only to the measurement scale used by the experimenter. In theory, the value of the mathematical expectation can be equal to any number from the range of numbers from -∞ to + ∞, and the variance to any nonnegative number. Therefore, there are an infinite number of different types of normal distribution and, accordingly, an infinite set of curves representing it (having, however, a similar bell-shaped shape). It is clear that all of them can not be described. However, if the parameters of a particular normal distribution are known, it can be converted to the so-called single normal distribution, whose mathematical expectation is zero and the variance to unity. This normal distribution is also called standard or z-distribution. The graph of the unit normal distribution is shown in Fig. 1.1, whence it is obvious that the vertex of the bell curve of the normal distribution characterizes the value of the mathematical expectation. Another parameter of the normal distribution - variance - characterizes the degree of "spreading" bell-shaped curve relative to the horizontal (abscissa axis).
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