X
is
distributed uniformly
on interval (
a
,
b
) if all its possible values are on this interval and its density function is constant:
(29)
For random variable
(30)
Fig. 4. Graph of uniform distribution density function
Examples of uniformly distributed values are mistakes of rounding up. So, if all tabular values of some function are approximated to the same digit then choosing at random tabular value, we consider, that mistake of rounding up of chosen number is a random variable uniformly distributed in interval
X
has an
exponent distribution
if its density function is expressed by the formula:
(31) Graph of density function (31) is represented on Fig. 5.
Fig. 5. Graph of exponent distribution density function
Time
X
has
normal (Gaussian) distribution
if density function is defined by the dependence:
(32)
where
At
Graph of normal distribution density function (32) is represented on Fig. 6.
Fig. 6. 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 addendum s, are distributed practically according to normal law.
Random variable
(33) where – Euler’s gamma function. The basic properties of gamma function :
Parameters
– any positive numbers. Gamma distribution is also
Pearson’s distribution of type III
[3]. At
gamma distribution turns to exponent distribution with parameter
Fig. 7. Graphs of gamma distribution density functions |

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>> Elements of Mathematical Statistics
>> Distributions of continuous random variables