DEVOTED TO 210-YEARS ANNIVERSARY OF KHARKIV NATIONAL MEDICAL UNIVERSITY

MEDICAL AND BIOLOGICAL PHYSICS

Lectures

MINISTRY OF PUBLIC HEALTH OF UKRAINEKHARKIV NATIONAL MEDICAL UNIVERSITY

DEPARTMENT OF MEDICAL AND BIOLOGICAL PHYSICS AND MEDICAL INFORMATICS

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 PharmacyAuthors:

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

RANDOM VARIABLES

Kharkiv National Medical University

Department of medical and biological physics and medical informatics

Plan Plan of of the the lecturelecture

1. Random variable (definition, types)

2. Discrete random variable; the distribution law; the condition of normalization

3. Continuous random variable; the probability density function; the condition of normalization; the distribution function

4. Numerical characteristic of random variable

5. Binomial distribution (Bernoulli distribution)

6. Normal distribution (Gauss distribution)

1!0

n...21!n

qp)!mn(!m

!nP mnm

b

adxf(X)=b)XP(a

1=pn

1=i i

•A random variable is a variable quantity that A random variable is a variable quantity that randomly assumes a certain numerical value from a randomly assumes a certain numerical value from a set of possible values resulting from a trial.set of possible values resulting from a trial.

•The occurrence of any value of this variable is a random event.

•There are discrete and continuous random variables.

•A random variable is called a discrete random A random variable is called a discrete random variable if it has a finite or countable set of possible variable if it has a finite or countable set of possible events. 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 enumerate all possible values of this variable and all possible values of this variable and indicate the probabilities, which these values indicate the probabilities, which these values are assumedare assumed.

•The law of a discrete random variable distribution is shown in the following table:

xi x1 x2 ... xn

p(xi) p(x1) p(x2) ... p(xn)

Values of X

Probabilities p(xi)

• 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

• The latter formula is called the condition of condition of normalisation of a discrete random variablenormalisation of a discrete random variable.

1=p...ppp=p n

n

1=i 321i

A continuous random variableA 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 Taking into account that a continuous random variable assumes an infinite set of values, the variable assumes an infinite set of values, the probability of the event that it will assume a certain probability of the event that it will assume a certain concrete value equals zero.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 functionthe probability density function (or the frequency functionthe frequency function) of a continuous random variable X, i.e.

dx

dP=f(X)

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 non-negative 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 the probability of values of variable X falling in the interval from a to binterval from a to b is calculated using formula

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:

,dxf(X)=b)XP(ab

a

The event consisting in that a random variable The event consisting in that a random variable will take any value in the interval from -∞ to +∞ will take any value in the interval from -∞ to +∞ is certain.is certain.

Therefore,

1)X P(- Hence,

+

-1dxf(X)

This formula is called the condition of This formula is called the condition of normalisation for a continuous random value.normalisation for a continuous random value.

To specify a continuous random variable, besides using the probability density, one can use the distribution functionthe 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

-dx,f(X)=F(X)

dx

dF(X)=f(X)

• 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 The probability of variable X falling in the intervalinterval from a to b is calculated with the distribution function F(x) using the formula

P(a X b) = F(b) - F(a)

•Among the numeral characteristics of random variable X we shall consider three of them:

mathematical expectation М(Х),mathematical expectation М(Х),

variance D(X), variance D(X),

standard deviation standard deviation (Х).(Х).

Numeral Characteristics of Random Variables

• The notion of mathematical expectation of random variable The notion of mathematical expectation of random variable XX almost coincides with the notion of the mean value of this almost coincides with the notion of the mean value of this variable.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:

)P(xx++)P(xx+)P(xx=)P(xx=M(X) nnn

1=i2211ii

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:

dxf(X)x=M(X)

•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.

2M(X)XM=D(X) If X is a discrete random variable, its variance is found by the If X is a discrete random variable, its variance is found by the

formulaformula

n

1=ii

2i )P(xM(X))(x=D(X)

If X is a continuous random variable, the variance is found by If X is a continuous random variable, the variance is found by the formulathe formula

(X)dxfM(X))(x=D(X) 2

In practice, for calculating variance the following formula is used more often

22 (M(X)))M(X=D(X)

n

1=ii

2i

2 )P(xx=)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

XX is a discrete random variable is a discrete random variable;

and if and if XX is a continuous random variable then is a continuous random variable then

+ 22 f(X)dxx=)M(X

)X(D)X(

The standard deviation is the The standard deviation is the square root of its variance, square root of its variance,

i.e.i.e.

• 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 the Bernoulli formulaBernoulli formula:

Binomial Distribution (Bernoulli distribution)

mnmmn

mnm qpCqp)!mn(!m

!n)m(P

The right side of the Bernoulli formula is the common term of Newton's binomial expansion

Therefore, the distribution of discrete random variable,

wherein the probability of each value is equal to

n

0m

mnmmn

n qpC)qp(

is called the law of binomial probability distributionthe law of binomial probability distribution.The following table can present the distribution:

Х 0 … m … n

Р(Х) … …nq mnmmn qpC np

,)qp( n

The mathematical expectationThe mathematical expectation and variance of discrete random variable Xvariance of discrete random variable X having a binomial distributiona binomial distribution are calculated by the following formulas:

М(Х) = np D(X) = npq

The importance of studying the normal distribution is The importance of studying the normal distribution is connectedconnected, 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.

Normal Distribution (Gauss Distribution)

If a continuous random variable has a normal distribution, then its probability density function is described by the formula

where аа=M(X)=M(X) is is the mathematical expectationthe mathematical expectation of variable X, and ==(X)(X) is the standard deviationthe standard deviation of variable X.

2

2

2

)ax(

e2

1=(X)f

A normal distribution graph has a bell-shaped form.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 the graph shifts along the X–axisshifts along the X–axis without changing its shapewithout changing its shape.

If If decreases while a is constant decreases while a is constant, then the graph the graph compresses to straight-line compresses to straight-line хх = = аа..

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 functionthe Laplace function φ(t) of the form

t

0

2

t

dte2

1)t(

2

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

where

2

1

x

x1221 )(t)(t dxf(X)=)xXP(x

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.

;ax

t 11

axt 22

• When finding the values of the Laplace function using the table, bear in mind that the Laplace function is an odd the Laplace function is an odd functionfunction, i.e.

(-t) = - (-t) = - (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 the Laplace functionLaplace function:

F (x) = 0.5 + F (x) = 0.5 + (t)(t)

Thank You for Attention!

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