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DEVOTED TO 210-YEARS ANNIVERSARY OF KHARKIV NATIONAL MEDICAL UNIVERSITY
MINISTRY OF PUBLIC HEALTH OF UKRAINE
KHARKIV NATIONAL MEDICAL UNIVERSITY
DEPARTMENT OF MEDICAL AND BIOLOGICAL PHYSICS AND MEDICAL
INFORMATICS
MEDICAL AND BIOLOGICAL
PHYSICS
Lectures
KHARKIV - 2014
УДК 61:53+577.3](07.07)
ББК 28.901я7
М42
Approved by the Academic Council of Kharkiv National Medical University (minute N 5 at 22.05.2014)
Reviewers:
Berest V.P. - associate professor of Department of Molecular and Medical biophysics, PhD (Math.
and Physics), V. N. Karazin Kharkiv National University
Timanyuk V.O. - Chief of Department of Physics, professor, National University of Pharmacy
Authors:
Knigavko V.G., Zaytseva O.V., Batyuk L.V., Bondarenko M.A
M42 Medical and Biological Physics. Lectures (in 2 parts): Textbook for students
studying the subject in English: In 2 parts / Vladimir G. Knigavko, Olga V. Zaytseva, Lilia V. Batyuk,
Marina A. Bondarenko.-Kharkiv: Kh.N.M.U., 2014.; Part I - 337p., Part II - 254p.
The Textbook covers the most important topics of medical and biological physics in compliance with the typical
educational program. The structure and contents of the lectures completely correspond to credit-module system of
educational process organization.
The lectures are intended for teachers and students of the medical Universities, as well as for all interested in
medical and biological physics.
All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or
storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this
publication) without the written permission of the publishers.
УДК 61:53+577.3](07.07)
ББК 28.901я7
© Kharkiv National Medical University, 2014
Kharkiv National Medical
University
RANDOM VARIABLES
Department of medical and biological
physics and medical informatics
Plan of the lecture
n

i
i=1
1.
Random variable (definition, types)
2.
Discrete random variable; the
distribution law; the condition of
normalization
3.
Continuous random variable; the P(a  X  b) = b f(X)  dx
a
probability density function; the
condition of normalization; the
distribution function
4.
Numerical characteristic of random
variable
5.
Binomial distribution
distribution)
(Bernoulli
6.
Normal
distribution
distribution)
(Gauss
p =1
n!
p m q nm
m! ( n  m )!
n!  1  2  ... n
0!  1
P
•A random variable is a variable quantity that
randomly assumes a certain numerical value from a
set of possible values resulting from a trial.
•The occurrence of any value of this variable is a
random event.
•There are
variables.
discrete
and
continuous
random
•A random variable is called a discrete random
variable if it has a finite or countable set of possible
events.
•For example, the number of students attending a
lecture, the number of boys born at a maternity house in
one day.
•To obtain a representation of a discrete random
variable, it is necessary to specify the
distribution law of this variable, i.e. enumerate
all possible values of this variable and
indicate the probabilities, which these values
are assumed.
•The law of a discrete random variable
distribution is shown in the following table:
Values of X
xi
x1
x2
...
xn
Probabilities
p(xi)
p(xi)
p(x1)
p(x2)
...
p(xn)
• Such a table can contain a finite or infinite
number of columns.
• Events consisting in that any possible value of a
random variable resulting from a trial can occur
are exclusive and form a complete group of
events. Hence
n

i=1
pi = p1  p2  p3 ...  pn = 1
• The latter formula is called the condition of
normalisation of a discrete random variable.
A continuous random variable is a
random variable that can assume
any value belonging to an interval
(intervals)
where
it
exists.
For example, the temperature of a
person, the duration of human life, the
diameter of a pupil, the cardiac cycle
duration and the blood sugar content
are all examples of continuous random
variables.
•Taking into account that a continuous random
variable assumes an infinite set of values, the
probability of the event that it will assume a certain
concrete value equals zero. The probability of the
event that a continuous random variable will take a value
from a certain interval is not equal to zero.
•If we divide the domain of existence of a random
variable to a number of intervals, and for each of these
intervals define the probability of a random event falling
therein, then the more intervals this domain would be
divided to, the more precise the variable would be.
•A continuous random variable would be defined
most precisely if the interval dimensions tend to
zero and the number of intervals tend to infinity.
The variable is equal to the ratio of the probability dP of a
random event falling in the interval from x to x+dx to the value
of this interval dx is called the probability density function (or
the frequency function) of a continuous random variable X, i.e.
dP
f(X) =
dx
where f(X) is the probability density function of a continuous
random variable X.
•Specifying the probability density function of a continuous
random variable is one of the ways of defining this function (i.e.
defining the law of distribution of this variable). From definition of
f(X) it follows that the probability density function is a nonnegative variable, i.e. f(X) 0.
Knowing the probability density function of variable X, one can
calculate the probability of this variable falling in any interval.
Thus, if the probability density function of variable X equals f(X),
then the probability of values of variable X falling in the
interval from a to b is calculated using formula
P(a  X
b
 b) = 
a
f(X)  dx ,
i.e. it is equal to the area of the curvilinear trapezium S under the
f(X) curve in the interval from a to b:
The event consisting in that a random variable
will take any value in the interval from -∞ to
+∞ is certain.
Therefore,
P(-   X   )  1
Hence,
+
 f(X)  dx
-
1
This formula is called the condition of
normalisation for a continuous random value.
To specify a continuous random variable,
besides using the probability density, one
can use the distribution function. The
distribution function F(X) of a continuous
random variable X is related to the
probability density f(X) of this random
variable by the formulas
x
F(X) =  f(X)  dx,
-
dF(X)
f(X) =
dx
• We can see from this formula that the
distribution function is equal to the probability of
the random variable assuming a value in the
interval from  tо x, or, in other words, that it will
take a value less or equal to x.
• With increasing x, the distribution function
increases or remains constant, the codomain of
distribution function being 0  F(X)  1.
• The probability of variable X falling in the
interval from a to b is calculated with the
distribution function F(x) using the formula
P(a  X  b) = F(b) - F(a)
Numeral Characteristics of Random
Variables
•Among the numeral characteristics
of random variable X we shall
consider three of them:
mathematical expectation М(Х),
variance D(X),
standard deviation (Х).
• The notion of mathematical expectation of random variable
X almost coincides with the notion of the mean value of this
variable.
•The relation of these notions will be described at length when
studying mathematical statistics.
•The mathematical expectation of a discrete random variable
is calculated by the formula:
n
M(X)=  xi  P(xi )= x1  P(x1 )+x2  P(x 2 ) ++ xn  P(x n )
i=1
here x1, x2,…,xn are all possible values of variable X, аnd P(x1),
P(x2),…,P(xn) are their respective probabilities.
•For calculating M(X), when X is a continuous random
variable, the following formula is used:
M(X) =

 x

f(X)  dx
•Variance and standard deviation characterise the magnitude of
deviation (spread) of values of a random variable from its
mathematical expectation.
•Variance of random variable X is the mathematical
expectation of the standard deviation of the values of this
variable from its mathematical expectation, i.e.
2
D(X) = M  X  M(X)
If X is a discrete random variable, its variance is found by the
formula
n
D(X) =  (xi  M(X))2  P(xi )
i=1
If X is a continuous random variable, the variance is found by
the formula

2
D(X) =  (x  M(X)) 

f (X)dx
In practice, for calculating variance the following formula
is used more often
2
2
D(X) = M(X )  (M(X))
i.e. the variance of random variable X is the difference
between the mathematical expectation of random
variable X squared and the square of its mathematical
expectation. At this, M(X2) is calculated by the formula, if
X is a discrete random variable;
M(X
2
n 2
) =  xi
i=1
 P(xi )
and if X is a continuous random variable then
+
2
2
M(X ) =  x

 f(X)dx
The standard deviation is the
square root of its variance,
i.e.
 ( X )  D( X )
Binomial Distribution (Bernoulli distribution)
• The binomial distribution (or Bernoulli distribution) is
one of the kinds of discrete random variable
distributions.
• Let random variable X be the number of event A
occurrences in n repeated independent trials. Also let
the probability of event A occurrence in each trial equal
p, and the probability of non-occurrence in each trial be
q, thereat q=1-р. Then the probabilities of values of
random variable Х (0, 1,…, m,…, n) can be found by the
Bernoulli formula:
n!
m nm
m m nm
P( m ) 
p q
 Cn p q
m! ( n  m )!
The right side of the Bernoulli formula is the common term of
Newton's binomial expansion
( pq) 
n
n
m m nm
 Cn p q
m0
Therefore, the distribution of discrete random variable,
wherein the probability of each value is equal to ( p  q ) ,
n
is called the law of binomial probability distribution.
The following table can present the distribution:
Х
0
…
m
…
Р(Х)
n
…
m m nm
Cn p q
…
q
n
p
n
The mathematical expectation and
variance of discrete random variable X
having a binomial distribution are
calculated by the following formulas:
М(Х) = np
D(X) = npq
Normal Distribution (Gauss Distribution)
The importance of studying the normal distribution is
connected, in particular, with the fact that many variables,
which characterise certain biological and medical objects, have
distribution laws that are very close to the normal law.
Such distribution laws are found in the following:
- the height and weight of adults;
- arterial blood pressure when examining a great number of
patients;
- the length of blood vessels; volume of organs; the weight and
volume of brains found when performing anatomic
examinations;
- the absolute errors of readings of instruments, and
measurement values;
- the enzyme content in healthy people.
If a continuous random variable has a normal
distribution, then its probability density function is
described by the formula
1
f (X) =
e
 2

2
( x a )
2
2
where а=M(X) is the mathematical expectation of
variable X, and =(X) is the standard deviation of
variable X.
A normal distribution graph has a bell-shaped form.
It is symmetrical with respect to straight-line.
If we change a while  is constant, then the graph
shifts along the X–axis without changing its shape.
If  decreases while a is constant, then the graph
compresses to straight-line х = а.
The area under it is always equal to 1.
The Laplace function
To calculate the probability of normally distributed
variable X falling in a certain interval it is necessary to
integrate the above expression for f(X). This integral
cannot be expressed through elementary functions. It is
calculated by the Laplace function φ(t) of the form
2
1
( t ) 
2
t
t 
2
e

dt
0
The Laplace function values have been tabulated using
numerical methods.
If random variable X has a normal distribution, the
probability of its falling in the interval [x1, x2] equals
P(x1  X  x2 ) =
where
t1 
x1  a

;
x2

x1
f(X)  dx  (t 2 ) (t1 )
t2 
x2  a

Thus, calculating the probability of a normally distributed
random variable falling in the interval [x1, x2] is reduced
to defining values t1 and t2, and finding the values of the
Laplace function (t1) and (t2) in the table.
• When finding the values of the Laplace
function using the table, bear in mind that
the Laplace function is an odd
function, i.e.
(-t) = - (t)
• The distribution function of a normally
distributed random variable also cannot
be
expressed
through
elementary
functions, but it can be expressed by the
Laplace function:
F (x) = 0.5 + (t)
Thank You for Attention!
Literature
1. Биофизика: учебник для вузов / П.Г. Костюк, Д.М. Гродзинський, В.Л. Зима и др.; Под общ. ред. П.Г. Костюка .- К.: Вища шк., 1988. 504 с.
2. Биофизика / Ю.А. Владимиров, Д.И. Рощупкин, А.Я. Потапенко, А.И. Деев.– М.: Медицина, 1983. – 272 с.
3. Волькенштейн М.В. Биофизика. – М. : Наука, 1988. - 590 с.
4. Вибрационная безопасность. Общие требования. ГОСТ 12.1.012-90. – М., 1990.
5. Гамалея Н.Ф., Рудых З.М., Стадник В.Я. Лазеры в медицине. – К.: Здоровье, 1988. – 43 с.
6. Гродзинський Д.М. Радіобіологя. – К.: Либідь, 2000. – 448 с.
7. Губанов В.И., Утепбергенов А.А. Медицинская биофизика. – М.: Медицина, 1978. – 336 с.
8. Ємчик Л.Ф., Кміт Я.М. Медична і біологічна фізика. – Львів.: Світ, 2003. - 592 с.
9. Кольченко В.В., Паничкин Ю.В. Ультразвук и сердце. – К.: Здоровья, 1988. – 45 с.
10. Кортуков Е.В., Воеводский В.С., Павлов Ю.К. Основы материаловедения. – М.: Высшая школа, 1988.– 322 с.
11. Котык А., Яначек К. Мембранный транспорт. Междисциплинарный подход. – М.: Мир, 1980. – 341 с.
12. Луизов А.В. Физика зрения. – М., Знание, 1976. – 62 с.
13. Маршелл Э. Биофизическая химия. Принципы, техника и приложения. В 2-х томах. – М.: Мир, 1981. – 824 с.
14. Мэрион Дж. Общая физика с биологическими примерами. – М.: Высшая школа, 1986. – 623 с.
15. Медична і біологічна фзика.Том 1. / О.В.Чалий, Б.Т.Агапов, А.В. Меленевська та ін. – К.: ВІПОЛ, 1999. – 425 с.
16. Медична і біологічна фзика. Том 2. / О.В.Чалий, Б.Т.Агапов, А.В. Меленевська та ін. – К.: ВІПОЛ, 2001. – 415 с.
17. Проблемы прочности в биомеханике / И.Ф.Образцов, И.С.Адамович, А.С.Барер и др. – М.: Высшая школа, 1988. – 311 с.
18. Рего К.Г. Метрологическая обработка результатов технических измерений. – К.: Техніка, 1987. – 128 с.
19. Ремизов А.Н. Медицинская и биологическая физика. – М.: Высшая школа, 1999. – 616 с.
20. Рубин А.Б. Биофизика / Учеб. пособие для биол. спец. вузов. – М.: Высш. шк., 1987. – 319 с. (1 том) 302 с (2 том)
21. Рубин А.Б. Термодинамика биологических процессов / Учеб. пособие для вузов. – М.: Изд-во МГУ, 1984. – 285 с.
22. Стенли А. Гельфанд. Слух: введение в психологическую и физиологическую акустику. – М.: Медицина, 1984. – 352 с.
23. Тиманюк В.А., Животова Е.Н. Биофизика. – Харьков: Изд-во НФАУ; Золотые страницы, 2003. – 704 с.
24. Ультразвук. Маленькая энциклопедия. Глав. ред. И.П.Голямина. – М.: Советская Энциклопедия, 1979. – 400 с.
25. Ультразвук. Общие требования безопасности. ГОСТ 12.1.001-89. – М., 1989.
26. Физика визуализации изображений в медицине: В 2-х т. Том 1; пер. с англ./Под ред. С.Уэбба. – М.: Мир, 1991. – 408 с.
27. Физика визуализации изображений в медицине: В 2-х т. Том 2; пер. с англ./Под ред. С.Уэбба. – М.: Мир, 1991. – 423 с.
28. Фотометрия. Термины и определения. ГОСТ 26148-84. – М., 1984.
29. Хауссер К.Х., Кальбитцер Х.Р. ЯМР в медицине и биологии: структура молекул, томография, спектроскопия in-vivo. – К.: Наук.
Думка, 1993. – 259 с.
30. Холл Э. Дж. Радиация и жизнь. – М.: Медицина, 1989. – 256 с.
31. Шандала М.Г., Думанский Ю.Д., Иванов Д.С. Санитарный надзор за источниками электромагнитных излучений в окружающей
среде. – К.: Здоровье, 1990. – 150 с.
32. Шум. Допустимые уровни в жилых и общественных зданиях. ГОСТ 12.1.036-81. – М., 1981.
33. Электростатические поля. Допустимые уровни на рабочих местах и требовании к проведению контроля. ГОСТ 12.1.045-84. – М.,
1984.
34. Ярмоненко С.П. Радиобиология человека и животных. - М.: Высшая школа, 1988. – 423 с.