1. tch-prob1 Chap 3. Random Variables The outcome of a random experiment need not be a number. However, we are usually interested in some measurement or numeric attribute of the outcome. For example, toss a coin n times, total number of heads = ? What is the prob. of the resulting numerical values ? A random variable X is a function that assigns a real number,, to each outcome.

2. tch-prob2 Ex.3.1 : HHH HHT HTH THH HTT THT TTH TTT : 3 2 2 2 1 1 1 0 Let B be some subset of Sx Event B in Sx occurs whenever event A in S occurs. Events A and B are equivalent events.

3. tch-prob3 HHH HHT HTH THH HTT THT TTH TTT 0 1 2 3

4. tch-prob4 3.2 Cumulative Distribution Function Cumulative distribution function (cdf) of a random variable X is defined as a convenient way of specifying the probability of all semi-infinite intervals of the real line of the form.

5. tch-prob5 Cdf has the following properties: i. cdf is a prob., axiom 1 & corollary 2. ii. is the entire sample space & Axiom II. iii. is an empty set. Corollary 3. iv. nondecreasing function, corollary 7. v. continuous from the right. h>0

6. tch-prob6 vi. vii. if cdf is continuous at b. viii. Corollary 1.

7. tch-prob7 Ex.3.4. Fig 3.3 toss a coin 3 times. Count Heads. probability mass function (pmf) For discrete random variable, cdf is right-continuous, staircase function of x.

8. tch-prob8 Ex.3.5. The transmission time X of messages obeys the exponential probability law with parameter. is continuous for all x, its derivative exists everywhere except at x=0. pdf.

9. tch-prob9 Continuous random variable cdf is continuous everywhere, and smooth enough. can be written as an integral of some nonnegative function f(x) property (vii)

10. tch-prob10 Random variable of mixed type Ex. 3.6 The waiting time X of a customer in a queueing system is zero if he finds the system idle (p), and an exponentially distributed random length of time if he finds the system busy (prob. 1-p).

11. tch-prob11 3.3 Probability Density Function (pdf) The pdf of X, if it exists, is defined as density

12. tch-prob12 i. ii. iii. iv. since is nondecreasing pdf completely specifies the behavior of continuous random variables.

13. tch-prob13 Ex. 3.7 uniform random variable ab 1/(b-a)

14. tch-prob14 The derivative of the cdf does not exist at points where the cdf is not continuous. To generalize pdf for discrete random variable. Define delta function

15. tch-prob15 Ex

16. tch-prob16 Conditional cdf’s and pdf’s The conditional cdf of X given A is satisfies all the properties of a cdf. The conditional pdf. of X given A is

17. tch-prob17 Ex.3.10 The lifetime X of a machine has a continuous cdf. Find the conditional cdf and pdf given A={X>t}.

18. tch-prob18 3.4 Some important random variables - Discrete Random Variables 1. Bernoulli r.v. 2. Binomial Random variable X: number of times a certain event occurs in n independent trials. Indicator function for A

19. tch-prob19 3. Geometric r.v. M indep. Bernoulli trials until the first success or M’=M-1, number of failures before a success the only discrete r.v. that satisfies the memoryless property:

20. tch-prob20 4. Poisson r.v. counting the number of occurrences of an event in a time period. average number of event occurrences in a time interval t. Figure 3.10.

21. tch-prob21 Binomial prob. Poisson prob. As

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23. tch-prob23 - Continuous r.v. 1. uniform r.v. 2. Exponential r.v. model the time between event occurrences.

24. tch-prob24 Exponential r.v. is limiting form of the geometric r.v. - An interval of duration T is divided into subintervals of length - Perform a Bernoulli trial on each subinterval with prob. of success - The number of subintervals until the occurrence of a successful event is a geometric r.v. M. - Thus, the time until the occurrence of the first successful event is X=M (T/n) 0 T

25. tch-prob25 For a Poisson r.v., the time between events is an exponentially distributed r.v. with parameter events/sec. Exponential r.v. also has the memoryless property The probability of having to wait at least h additional seconds given that one has already been waiting t seconds = The probability of waiting at least h seconds when one first begin to wait.

26. tch-prob26 3. Gaussian (Normal) r.v. Sum of a large number of small r.v.s p.d.f cdf. change of variable where

27. tch-prob27 Ex. 3.14. Show that Gaussian pdf integrates to one.

28. tch-prob28 Ex.3.15. Q-function Table 3.3 It is sometimes convenient to work with Q(x).

29. tch-prob29 Q(x) x 00.500 1.00.159 2.02.28E-2 3.01.35E-3 4.03.17E-5 5.02.87E-7 6.09.87E-10 k 11.2815 22.3263 33.0902 43.7190 54.2649 64.7535 75.1993

30. tch-prob30 4. Gamma r.v. Pdf where is the gamma function m non negative integer m-Erlang r.v. exponential r.v. Figure 3.14

31. tch-prob31 ※ : the time until the occurrence of the mth event Assume the times between events are exponential r.v., (Poisson r.v. limiting case) Let N(t) be the Poisson r.v. for the number of events in t seconds. ※ iff m th event occurs before t m or more events occur in t second. m-Erlang

32. tch-prob32 3.5 Functions of a Random Variable X: r. v. g(x): real-valued function Y=g(X) is also a r.v. Event C in Y equivalent event B in X

33. tch-prob33 Ex 3.21. X: # of active speakers in a group of N indep. speakers p: Prob. that a speaker is active A Voice transmission system can transmit up to M voice signals at a time Y: # of Signals discarded

34. tch-prob34 Y=aX+b, where a is nonzero. Suppose X is continuous and has cdf, Find. pdf.

35. tch-prob35 Ex.3.24 X: Gaussian If has n solutions,, then will have n terms.

36. tch-prob36 Consider a nonlinear function Y=g(X) event Its equivalent event dx2 is negative Ex.3.27.

37. tch-prob37 Ex. 3.28 Y=cos(X), X: uniformly distributed in for –1< y <1, y has two solutions

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39. tch-prob39 3.6 The Expected Value of R.V.s c.d.f. or p.d.f provides complete description of a r.v. Sometimes interested in a few parameters that summarize the information Expected value of X or mean of X is defined by for discrete r.v. The expected value is defined if

40. tch-prob40 When the pdf is symmetric about a point m, i.e., If Ex..Gasussian uniform

41. tch-prob41 X continuous X integer-valued When X is a non-negative r.v. 1

42. tch-prob42 Ex. Exponential r.v. Expected value of Y=g(X)

43. tch-prob43 Ex.3.33 constant uniform r.v. in

44. tch-prob44 If c is a constant If

45. tch-prob45 Variance of X Expected value provides limited info., also want to know variation magnitude Ex. 3.38. Var. of Gaussian (x-m) dx

46. tch-prob46 Var [c]=0 Var [X+c]= Var [X] Var [cX]= nth moment of the random variable X

47. tch-prob47 3.7 Markov and Chebyshev Inequalities Suppose X is a non-negative r.v., Markov inequality 1 a

48. tch-prob48 Suppose are known. Let and use Markov inequality, we obtain Chebyshev inequality

49. tch-prob49 Ex. 3.42 Suppose Then the Chebeshev inequality for gives Now suppose that we know that X is a Gaussian r.v., then for k=2

50. tch-prob50 3.9 Transform Methods useful computational aids in the solution of equations that involves derivatives and integrals of functions. A. Characteristic Function Expected value of Fourier Transform of Ex.3.47. Exponential r.v. check p.101.

51. tch-prob51 If X is a discrete r.v.. If X is a discrete integer-valued r.v., A periodic function of with period Fourier series coefficients of

52. tch-prob52 If f(x) is a periodic function of period, then f(x) can be represented as

53. tch-prob53 Moments of X can be obtained from by

54. tch-prob54 Ex.3.49exponential Check p.101

55. tch-prob55 B. Probability Generating Function For nonnegative r.v. a. if N is nonnegative integer-valued r.v. prob. Generating Function of N

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57. tch-prob57 b. if X is a non-negative continuous r.v. Laplace transform of the Ex. 3.51. Laplace transform of the gamma

58. tch-prob58