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Topics of lectures

Lecture Notes Subject : Medical and Biological Physics Group : __________________ Name : __________________. Topics of lectures. 1. Random variables 2. Mathematical Statistics 3. Mechanics of fluids 4. Bioacoustics 5. Physical bases of electrography

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Topics of lectures

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  1. Lecture NotesSubject: Medical and Biological PhysicsGroup: __________________Name: __________________

  2. Topics of lectures 1. Random variables 2. Mathematical Statistics 3. Mechanics of fluids 4. Bioacoustics 5. Physical bases of electrography 6. Effect of electromagnetic fields on biological objects 7. Geometrical Optics. Lenses. Optical system of the human eye 8. The polarization of light 9. X-ray radiation 10. Dosimetry of ionizing radiation

  3. Lecture 1 THEORY OF PROBABILITIES.RANDOM VARIABLES

  4. Plan of the lecture • Basic notions of Probability Theory • Random variable (definition, types) • Discrete random variable; distribution law; condition of normalization • Continuous random variable; probability density function; condition of normalization; distribution function • Numerical characteristic of random variable • Binomial distribution (Bernoulli distribution) • Normal distribution (Gauss distribution)

  5. 1) Drawing a die (playing cube) is a trial, number “1” is RE, number “2” is RE, …, number “6” is RE as well Basic notions of Probability Theory • Theory of probability deals with random events. • Random event (RE) is event which may or may not occur when a certain set of conditions is realized. Every realization of such set of conditions is called a trial.

  6. 2)tossing a coin is a trial, “head” is a RE, “tail” is a RE 3) shooting the targetis atrial, hitting the aim is a RE 4) child-birth is a trial, boy-birth is a RE, girl-birth is a RE

  7. Outcome of the trial is an occurrence of any occasional result or event as a result of the trial. • If an expected event occurred as a result of the trial we have favourable outcome of the trial. • If the event did not occur an outcome is called unfavourable outcome.

  8. 0≤ m ≤ n =>0 ≤ P*(A) ≤ 1 • Denote a RE by A. • Suppose that the event A occurred m times in a series of n independent trials => m is a number of favourable outcomes, n is a number of trials. Relative frequency of the event A :

  9. Statistical probability of the random event A : • If the number of trials n is sufficiently large, the relative frequency of the event can be approximately taken as its probability. • This fact implies a method of empirical calculation of probability when it is difficult to find it theoretically.

  10. When there is no reason to suppose that any of the trial outcomes can be observed more or less often than others, such outcomes are called equally possible outcomes. • When tossing a die on whose faces number of points from 1 to 6 is marked there is no reason to suppose that the occurrence of any face (or side) can happen more or less frequent than any other side, e.g. occurrence of each side is one of the equally possible outcomes.

  11. Classical probabilityof random event A (formula is used if all trial outcomes are equally possible outcomes) n is a number of favourable outcomes, m is a general number of all equally possible outcomes

  12. Basic properties of Probability : • The probability of RE is dimensionlessquantity: 0 < P(A) < 1 2. The probability of a certain (sure) event is equal to unity: P(A)=1 3. The probability of an impossible eventis equal to zero: P(A)=0

  13. Random variable (RV)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. Two types of RV: • discrete random variable (DRV) • continuous random variable (CRV) Discrete random variable (DRV)is a random variable which has a finite or countable set of possible events. For example, number of students attending a lecture, number of boys born at a maternity house in one day.

  14. To represent DRV it is necessary to specify the distribution lawof this variable. Distribution law is a table, in which all possible values of DRV and all corresponding probabilities of these values are enumerated.

  15. 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. • Condition of normalisation for DRV:

  16. Continuous random variable (CRV)is a random variable that can assume any value belonging to an interval (intervals) where it exists.For example,temperature, height, weight, length,time, concentration – or duration of human life, cardiac cycle duration, diameter of a pupil, blood sugar content

  17. CRV assumes an infinite set of values • Probability of the event that CRV will assume a certain concrete value equals zero P(concrete value) = 0 The probability of the event that CRV will take a value from an interval is not equal to zero. Two ways to represent CRV: 1. probability density function (frequency function) f(X) 2. distribution functionF(X)

  18. where dP is probability of CRV to fall in interval from x to x+dx • Probability density function f(X) is a non-negative quantity : f(X)0 Probability density function(frequency function) f(X) :

  19. P(a≤X≤b) is equal to area S of the curvilinear trapezium under thef(X) curve in the interval from a to b: Probability of CRV X to fall in interval fromatob :

  20. The event consisting in that CRV will take any value in the interval from -∞ to +∞ is a certain event. Condition of normalisation for CRV :

  21. Distribution functionF(X) :

  22. With increasing x, F(X) increases or remains constant. Distribution function takes values: 0 F(X)  1 • Probability of CRV X to fall in interval from a to b : P(a  X  b) = F(b) - F(a)

  23. Numerical Characteristics of RV: 1. Mathematical expectation М(Х) 2. Variance D(X) 3. Standard deviation(Х)

  24. x1, x2,…,xnare all possible values of variableX, P(x1), P(x2),…,P(xn)are corresponding probabilities. Mathematical expectation for CRV: • The notion of mathematical expectation of RV X almost coincides with the notion of themean valueof this RV. Mathematical expectation for DRV:

  25. For DRV: For CRV: • Variance D(X) and standard deviation characterise the magnitude of deviation (spread) of values of RV from its mathematical expectation.

  26. Mathematical expectation of random variable X squared M(X2) : for DRV: for CRV: In practice :

  27. Standard deviation :

  28. Binomial Distribution (Bernoulli distribution) • The binomial distribution (or Bernoulli distribution) is one of the tipess of DRV-distributions. • Let random variable Xbe the number of event A occurrences in n repeated independent trials. Also let the probability of event A occurrence in each trial equals p, and the probability of non-occurrence in each trial be q, thereat q=1-р. Then the probabilities of values of DRV Х (0, 1,…, m,…, n) can be found by the Bernoulli formula:

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

  30. The mathematical expectation and variance of DRV X having a binomial distribution: М(Х) = np,D(X) = npq

  31. 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.

  32. If CRV has a normal distribution, then its probability density function f(X) is where а=M(X)is the mathematical expectationof variable X, =(X)isthe standard deviationof variable X.

  33. A normal distribution graph has a bell-shaped form. It is symmetrical with respect to straight-line x = a. If we change a while=const, then the graph shifts along the X–axiswithout changing its shape.Ifdecreases while a=const, then the graph compresses to straight-line х = а.Area S under the graph is always equal to one (S=1).

  34. 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 byLaplace functionφ(t) : Laplace function values have been tabulated using numerical methods.

  35. If CRV X has a normal distribution, the probability of its falling in the interval [x1, x2] equals where

  36. Laplace function is an odd function : (-t) = - (t) • Distribution function F(X)of a normally distributed CRV also can be expressed through Laplace function : F (x) = 0.5 + (t)

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