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Distribution laws of random variables

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Each law of distribution is some function completely describing random variable according to probable point of view. In practice often it is necessary to discuss about distribution of probabilities of random variable X only by results of tests. Repeating tests, each time let’s register, whether there was interesting us random event A or not. Relative frequency (or simply frequency) of random event A is called relation of number nA of occurrences of this event to general number n of executed tests. Thus we accept that relative frequencies of random events are close to their probabilities. It is especially true, than it is more number of executed experiences. Thus of frequencies, as well as probabilities, it is necessary to carry not to separate values of random variable, but to intervals. It means that all range of possible values of random variable X should be broken into intervals. Spending series of tests, giving empirical values of variable X, it is necessary to fix numbers nx of hits of results in each interval. At the big number of tests n the relation nx / n (frequencies of hit in intervals) should be close to probabilities of hit in these intervals. Dependence of frequencies nx / n from intervals defines empirical distribution of probabilities of random variable X which graphic representation refers to as histogram (Fig. 1).

Fig. 1. Histogram and aligning density function

 

For construction of histogram on x -axis intervals of equal length are set into which all range of possible values of random variable X is broken, and on y -axis frequencies nx / n are set. Then height of each column of histogram is equal to corresponding frequency. Thus, the approached representation of law of distribution of probabilities for random variable X in form of step function turns out, approximation (alignment) of which by some curve f (x) will give density function.

However, often happens to specify enough only separate numerical parameters describing basic properties of distribution. These numbers are called numerical characteristics of random variable.

 


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