2. Distribution laws of random variables

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 n_A 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 n_x of hits of results in each interval. At the big number of tests n the relation n_x / n (frequencies of hit in intervals) should be close to probabilities of hit in these intervals. Dependence of frequencies n_x / 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 n_x / 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.

Numerical characteristics of continuous random variables

Mathematical expectation. Mathematical expectation of discrete random variable X accepting finite number of values х_i with probabilities , is the sum:

(5a)   

Mathematical expectation of continuous random variable X is the integral from product of its values х and its density function f (x):

   (5b)  Not own integral (5) is supposed absolutely converging (otherwise, mathematical expectation M (X) does not exist).

Mathematical expectation characterizes average value of random variable X. Its dimension coincides with dimension of random variable.

Properties of mathematical expectation:

1) ;

2) ;

3) for independent random variables X and Y.       (6)

Dispersion. Dispersion of random variable X is the number:

.  (7)

The dispersion is characteristic of dissipation of random variable X values relatively its average value of M (X). Dimension of dispersion is equal to dimension of random variable square. Proceeding from definitions of dispersion (7) and mathematical expectation (4) for discrete random variable and (5) for continuous random variable we’ll receive similar expressions for dispersion:

Here m = М ( Х ).

Properties of dispersion:

1) ,

2) for independent random variables X and Y.      

 Average square-law deviation: .

As dimension of average square-law deviation the same, as at random variable, it is used as measure of dissipation more often, than dispersion.

Moments of distribution. Concepts of mathematical expectation and dispersion are special cases of more general concept for numerical characteristics of random variables – moments of distribution. Moments of distribution of random variable are entered as mathematical expectations of some elementary functions of random variable. So, the k order moment relatively point х₀ is called mathematical expectation M (X – х₀)^k. Moments relatively beginning of coordinates х = 0 are called initial moments and are designated:

Initial moment of the first order is centre value of distribution of considered random variable: .

Moments relatively centre value of distribution  х = m are called central moments and are designated: .

From (6) follows, that central moment of the first order is always equal to zero:

.

Central moments do not depend on reference mark of random variable values as at shift on constant value C its centre value of distribution moves on the same value C, and deviation from the center does not vary:  Х – m = (Х – С) – (m – С).

Mode of discrete random variable is its most probable value. Mode of continuous random variable is its value at which density function is maximal. If curve of distribution has one maximum, then its distribution refers to unimodal. If curve of distribution has more than one maximum, then its distribution refers to polymodal. Sometimes there are distributions, which curves have not maximum, but minimum. Such distributions refer to antimodal. Generally mode and mathematical expectation of random variable do not coincide. In that specific case, for modal, i.e. having mode, symmetric distribution and provided that there is mathematical expectation, it coincides with mode and center of distribution symmetry.

Median of random variable X is its value Ме for which the equality: takes place, i.e. equiprobably, that random variable  X  is less or more Ме. Vectorially median is abscissa of point in which area under curve of distribution halves. In case of symmetric modal distribution median, mode and mathematical expectation coincide.

Distributions of continuous random variables

Uniform distribution. Continuous random variable X is distributed uniformly on interval (a, b) if all its possible values are on this interval and its density function is constant:

For random variable X uniformly distributed in interval (a, b) (Fig. 2), probability of hit in any interval (x₁, x₂), laying inside of interval (a, b), is equal to:

.

Fig. 2. Graph of uniform distribution density function

 

Exponent distribution. Continuous random variable X has an exponent distribution if its density function is expressed by the formula:

(8)

Graph of density function (8) is represented on Fig. 3.

Fig. 3. Graph of exponent distribution density function

 

Time Т of non-failure operation of computer system is a random variable having exponent distribution with parameterλ, which physical sense – an average number of refusals in unit of time, not including idle times of system for repair.

Normal (Gaussian) distribution. Random variable X has normal (Gaussian) distribution if density function is defined by the dependence:

, (9)

where m = M (X) , .

At    normal distribution is called standard.

Graph of normal distribution density function (9) is represented on Fig.4.

 

Fig. 4. Graph of normal distribution density function

 

Normal distribution is the most often meeting in various casual natural phenomena. So, mistakes of performance of commands by automated device, mistakes of conclusion of spacecraft in the given point of space, mistakes of parameters of computer systems, etc. in the most cases have normal or close to normal distribution. Moreover, random variables formed by summing of great quantity of random addendums, are distributed practically according to normal law.

Gamma distribution.

Random variable X has gamma distribution if its density function is expressed by the formula:

where      – Euler’s gamma function. 

The basic properties of gamma function:

1) ;

2) , hence for any whole n>0 ;

3) , hence for any whole n>0

.

 Parameters – any positive numbers. Gamma distribution is also Pearson’s distribution of type III . At gamma distribution turns to exponent distribution with parameter λ, as Г (1) = 1. Gamma distribution is widely used in mathematical statistics. Graphs of gamma distribution density functions at are presented on Fig. 5.

Fig. 5. Graphs of gamma distribution density functions