Lecture 15. Elements of Mathematical Statistics. Random variables, their types. Distribution laws of random variables
1. Random variables, their types
The mathematical statistics is the science, engaged methods of processing of experimental data. Any science solves in ascending order complexity and importance the following problems:
1) phenomenon description;
2) analysis and forecast;
3) search of optimum decision.
Such problems are solved also with mathematical statistics:
1) to systematize the received statistical material;
2) on a basis of the received experimental data to estimate numerical characteristics of an observable random variable interesting us;
3) to define number of experiences, sufficient for reception of authentic results at the minimal mistakes of measurement.
One of problems of the third type is the problem of check of credibility of hypotheses. It can be formulated as follows: there is a set of the skilled data concerning one or several random variables. It is necessary to define, whether these data contradict this or that hypothesis, for example, to a hypothesis that the investigated random variable is distributed according to the certain law, or two random variables are not correlated (i.e. are not connected among themselves), etc. As a result of check of credibility of hypothesis either it is rejected, as contradicting skilled data, or is accepted as comprehensible.
Thus, mathematical statistics helps experimenter to understand better the received skilled data, to estimate, are significant or not the certain observed facts, to accept or reject those or other hypotheses about the nature of the considered phenomenon.
A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point.
For
example define the random variable X
as the number of heads in 2 tosses of a fair, 50-50 coin. The sample
space is
the corresponding outcomes in this sample space get associated with
values of the random variable X
as
because the outcomes have 1,2,1, and 0
heads respectively.
Random variables
Variable is called random if as result of experience it can accept valid values with certain probabilities. The fullest, exhaustive characteristic of random variable is law of distribution. Law of distribution is function (given by table, graph or formula), allowing to define probability of that random variable X accepts certain value хi or gets in some interval. If random variable has given law of distribution speak, that it is distributed under this law or submits to this law of distribution.
Random Variable:
Discrete random variable
Continuous random variable
A discrete random variable X has a finite number of possible values.
Random variable X is called discrete if there is such non-negative function
(1)
which puts in conformity to value хi of variable X probability рi with which it accepts this value.
Random variable X is called continuous if for any a < b there is such non-negative function
f (x),
that
(2)
Function f (x) is called density function of continuous random variable.
Probability of that random variable X (discrete or continuous) accepts value, smaller х, is called distribution function of random variable X and is designated F (x):
(3)
Distribution function is universal kind of distribution law, suitable for any random variable.
Basic properties of distribution function:
1)
-
not decreasing function, i.e. at
;
2)
- limited function:
;
3)
;
4)
;
5)
-
continuous on the left function.
Except for this universal, there are also private
kinds of distribution laws: series of
distribution
(only
for discrete random variables) and density
function
(only
for continuous random variables).
Basic properties of density function:
1)
;
2)
.
(4)
Random
variables that can assume values corresponding to any of the points
contained in one or more intervals are called continuous.