2.1 Concept of a Random Variable

1. 2. Single Random VariableHow to describe the behavior of a random variable?It is only possible for its moments.To derive the moments, pdf is necessary.So, various kind of pdf will be introduced in this Chapter.

2. 2.1 Concept of a Random Variable A random variable is a numerical description of the outcome of a random experiment. A random variable is a real valued function defined over the sample space S. outcome r.v

3. Example The actual value of resistance all marked “100Ω”, but lie between 9.99Ω and 100.01Ω is a random variable that can assume any value in a specified range of values. There are infinite number of such values in this range →continuous r.v.

4. Random Variables associated with Random Processes • Ensemble : the collection of all possible random time functions with probability functions specified • … a sample function • The value of the sample function as some particular time is a random variable • (Remarks) - There is a different random variable for each time instant - The randomness we are concerned with is the randomness between sample function and sample function - There maybe exists randomness from time instant to time instant, but this is not essential to random process - In summary, the probability description of random variables should be the probability description of random processes So, we will extend our discussion later to random processes

5. Random Variables associated with Random Processes

6. In this Chapter, we will discuss • What events are required for a complete description of the random variable? • How the appropriate probabilities can be inferred? • Probability distribution function

7. 2.2 Probability Distribution Functions2.3 Probability Density Functions # Probability distribution function: the probability of the event that the observed r.v. is less than equal to the allowed value # Properties of PDF i) ii) iii) Fx(x) is nondecreasing as x increases iv)

8. Although the probability distribution function is a complete description of the probability model for a single random variable,it is not the most convenient form for many calculations._________________________________________________ # Probability density function : # physical significance for infinitesimally small

9. Properties of pdf i) ii) iii) iv)

10. Transformation of Variables_________________________________ Given fX , find fY subject to

11. illustration

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14. 2.4 Mean Values and Moments # find average values of rv or functions of rv’s  ensemble averaging : integrating over the range of possible values that the rv may assume  the expected value of X # the expected value of any function of x

15. general moments & central moments # general moments of rv : n=1 : the mean value n=2 : the mean-square value # central moments of rv : n=1 : zero n=2 : the variance

16. (example)

17. 2.5 The Gaussian Random Variable Important because • A good mathematical model for many random phenomena • Can be extended to handle an arbitrarily large # of rv’s • Linear combinations of Gaussian rv’s lead to Gaussian • The random process can be completely specified by the first and the second moment only • The Gaussian process is often the only one for which a complete statistical analysis can be done in linear/nonlinear situation See Figure 2-11 in p. 68 ; pdf & PDF

18. Some feature of pdf of Gaussian rv • Only one max occurring at the mean value • pdf is symmetrical about the mean value • 최대크기의 0.607 때의 폭 • 최대크기 에 반비례

19. The Gaussian PDF cannot be expressed in closed form but can be expressed in terms of tabulated functions !  tabulated usually zero mean & unity variance

20. Q-function과 의 관계; Q(x) = 1 –

21. (Remarks) The Q-function is useful in calculating the probability of events that occur very rarely. • See the example of an trigger circuit in pp. 71-72

22. One of the most useful properties of Gaussian rv : high-order central moments can be determined easily ( ) n - = x x 0 n odd = - s n 1 , 3 , 5 , , , ( n 1 ) n even = n but x ? not always simple

23. # Central limit theorem : The pdf for the sum of a large # of independent rv’s having the same pdf approaches a Gaussian density function as the number becomes large regardless of the pdf of the independent rv’s Σ Assume rv’s X1, X2,… Xn same mean value m and same variance Then the normalized sum becomes Gaussian with zero mean and a variance of regardless of true pdf of Xi’s n 1 ( ) Σ = - Y X m K n = k 1

24. 2.6 Density functions related to Gaussian # There are many other pdf ‘s related to or derived from Gaussian pdf (i) Distribution of power Voltage or current ; random varable The power dissipated in a resistor Find fw(w), the pdf of W

26. Probability distribution function is given by

27. PDF represented in terms of the Q-function

28. (ii) Rayleigh Distribution - the peak values (the envelope) of a random voltage or current having a Gaussian pdf - the envelope of the sum of many sine waves of different frequencies - the errors associated with the aiming Y Miss Distance R X X,Y; independent Gaussian Random variables 과녁

29. pdf for the total miss distance R mean, mean-square value & variance Not symmetrical

30. (Remark) The mean & variance depend on a single parameter and cannot be adjusted independently! Probability Distribution function

31. A numeric example : an aiming problem Y 1 feet radius X Bulls-eye 1 inch radius 1 feet = 12 inches

32. (iii) Maxwell Distribution pdf of the velocity of a molecule in a perfect gas Vz V each velocity component ~ where Vx Vy Boltzmann’s constant Kelvin unit of temperature

33. Mean, mean-square value & variance

34. The mean kinetic energy • For obtaining the probability distribution function, it is necessary to carry out integration numerically. For example, see the example in p. 82 Mean kinetic energy

35. (iv) Chi-Square DistributionLet where are independent Gaussian rv’s with mean 0 and variance 1 Then, the random variables X2 is said to have a Chi-square distribution with n degrees of freedom and pdf is given by

36. (cf) Gamma function is defined by # Chi-square can be considered as the generalization. - the power distribution : chi-square with n=1 - the Rayleigh distribution : the square of the miss distance (R2) is chi-square with n=2 - the Maxwell distribution : the square of the velocity (V2) is chi-square with n=3

37. Mean & Variance # the chi-square distribution arises in many signal detection problems

38. (v) Log-Normal Distribution # random variables that are defined as the logarithms of other random variables For example, consider the attenuation of the signal power In the transmission path A is very often close to being a Gaussian random variable. Then, how about the pdf of the power ratio? Transmission Path Wout Win

39. # log-normal random variable Mean & Variance

40. The probability distribution function cannot be expressed in the closed form, but should be calculated by numerical integration.

41. 2.7 Other Probability Density Functions (i) Uniform distribution • arises usually where there is no preferred value for the random variable, for example, • the unknown phase angle in a sinusoidal source • the time position of pulses in a periodic sequence of pulses • the quantization error in an analog-to-digital conversion • pdf • Mean • Variance

42. The probability distribution function • ex) error in A/D conversion

43. ex) the phase of the received sinusoidal signal

44. Application of the uniform pdf # generation of samples of random variables having other probability density functions Given a uniform-distributed X over (0, 1), generate a random variable Y with pdf FY(y) • Find a function g(x) such that Y=g(X) will have FY(y) The answer is as follows :

45. ex) generate samples of a rv having a Rayleigh pdf

46. (ii) Exponential & Related Distributions # pdf for the time interval between events - assumptions: average time interval between events is ( Then, the probability that an event will occur in a time interval is ) - problem : Suppose that an event has occurred at t0, determine the probability that the next event will occur at a random time lying between and

47. Solution : For given , the probability to be determined is Note that the probability that the event occurred in must be equal to the product of the probabilities of the independent events that the event did not occur between t0 and and the event did occur between and , that is, From this, we can obtain

48. PDF is given by pdf is given by Mean & Variance

49. 1년 안에 고장날 확률 4년 이상 버틸 확률 10년 이상 버틸 확률 4년에서 6년 사이에 고장날 확률 Illustrative Application of Exponential Distribution (ex 1) Suppose the component’s MTBF is 100 days. Determine the probability of completing 200 day mission without a component failure. (ex 2) Suppose MTTF of a traveling tube is 4 years. The actual lifetime T is a rv with exponential distribution. Determine the probability associated with any specified lifetime :

50. Mean Variance Erlang distribution (Remark) The random variable in the exponential distribution is the time interval between adjacent events. This can be generalized to make the random variable the time interval between any event and the kth following event. # Erlang random variable of order k