Fundamentals of Probability, z-transform, and Laplace Transform

1. Intro. to Probability, z-transform, and Laplace transform Cheng-Fu Chou

2. Outline • Probability system • Conditoinal prob. • Theorem of total prob. • Bayes’ theorem • z-transform • Laplace transform

3. Probability • Sample Space (S) which is a collection of objects. Each is a sample point. • A family of event S = {A, B, C, …} where a event is a set of sample points. • A Probability measure P is an assignment (mapping) of events defined on S into the set of real numbers. • P[A] = Probability of event A

4. Properties • 0  P[A]  1 • P[S] = 1 • If A and B are mutually exclusive, then P[A  B] = P[A] + P[B]

5. Notations • Event • A = {w: w satisfies the membership property for the event A} • Example • dice

6. Ac is the complement of A where { w: w not in A} • A B = {w: w in A or B or both} = Union • A B = {w: w in A and B} = Intersection • 0 = empty set

7. In general • Let A1, A2, …, An be events

8. Conditional Prob. • Cond. Prob. • Constraint sample space • Scale up

9. Ex. 1

10. Ex. 2

11. Statistical Independent

12. Bayes’ Theorem • Bayes’ Theorem: we can look at the problem from another perspective. Assume we know event B has occurred, but we want to find which mutually exclusive event has occurred.

13. Ex. 1 • Assume that 15% of job is from I.E., 35% of job is from E.E., 50% job is from C.S.I.E.. Probability of read news are 0.01, 0.05, and 0.02 respectively. • P[a job chosen in random is a Read new jobs] • P[a randomly chosen job is from EE | it is a read news job]

14. P[read] = 0.15*0.01+0.35*0.05+0.5*0.02 • P[read from E.E./read] = (0.35*0.05)/P[read]

15. Ex. 2 • When you walk into a casino, it is a equal prob. for you to play with honest player or a cheating player. The prob. for you to lose when playing with a cheating player is p. Find the • Prob.[playing with a cheating player | you lost]

16. P[playing with a cheating player / you lost] = 0.5*p/(0.5*0.5+0.5*p)

17. Bernoulli trials • A random experiment that has two outcomes: “success”: p or “failure” : q (= 1 - p). Now consider a compound sequence of n independent repetition of this experiments. This is known as Bernoulli trials. • What is the prob. of exactly k successes after n trials ? • Verification:

19. The Birthday Problem • In probability theory, the birthday paradox states that given a group of 23 (or more) randomly chosen people, the probability is more than 50% that at least two of them will have the same birthday. For 60 or more people, the probability is greater than 99%. • Prove them.

20. Birthday problem (cont.) • assume there are 365 days in a year • p is the prob. that no two people in a group of n people will share a common birthday • p = (1 – 1/365)(1 – 2/365) …(1 – (n-1)/365) • p < ½ as n is 23 • P < 0.01 as n is 56

21. Monty Hall Problem • Based on the American game show “Let’s Make a Deal” • Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

22. Random Variables (r.v.) • We have the probability system (S, S, P) • R.v. is a variable whose value depends upon the outcome of the random experiment. • The outcomes of our random experiment is a w  S. We associate a real number X(w) to w. • Thus, the r.v. X(w) is nothing more than a function defined on the sample space S.

23. Probability distribution function

24. Probability density function • pdf: • Different ways to view pdf:

25. Special discrete dist. • The Bernoulli pmf • The Binominal pmf: n independent bernoulli trials.

26. Geometric Dist. • Sequence of bernoulli trials, but we count no. of trials until First success • P(i) = P(x=i) = (1-p) i-1 p • For Geometric dist., it has markov property or it is memoryless: P[x = i + n | x >n } = P[ x = i]

27. Special Continuous Dist. • Exponential r.v. • A continuous r.v. for some l > 0 f(x) = l e –lx if x  0 0 x < 0 • For exponential dist., it has Markov property or it is memoryless

28. Question • How could we generate a sequence number x1, x2, …,xn such that xi is a exponential (or any other specific dist.) r.v. with parameter l?

29. Inverse-transform Technique • The concept • For cdf function: r = F(x) • Generate r from unifrom (0,1) • Find x such that x = F-1(r)

30. Ans

31. PDF for 2 R.V. • Marginal Density Function

32. Function of Random Variable • Func. of r.v. • One important r.v. is where Xi are independent

33. Y = X1 + X2 • PDF • pdf • Convolution

34. Convolution Ex. • g(n) • 1/2 as n = 1 • 1/2 as n = 2 • f(n) • 2/3 as n = 1 • 1/3 as n = 2 • h(n) = f(n)g(n) = ?

35. Ex. (cont.) • h(0) = f(0)g(0) = 0 • h(1) = f(1)g(0) + f(0)g(1) = 0 • h(2) = f(2)g(0) + f(1)g(1) + f(0)g(2) = 1/3 • h(3) = f(3)g(0)+f(2)g(1)+f(1)g(2)+f(0)g(3) = ½ • h(4) = f(4)g(0)+f(3)g(1)+f(2)g(2)+f(1)g(3)+f(0)g(4) = 1/6

36. z transform • Consider a func. Of discrete time fn s.t. • fn  0 for n = 0, 1, 2, … • fn = 0 for n = -1, -2, …

37. Examples • Ex1: • Ex2:

38. Convolution Property

39. Properties of z transform

40. Properties (cont.)

41. z Transform pair

42. z transform : difference equation • Ex:

43. z-transform and moment

44. Laplace transform • Def: • Ex 1. • Ex 2.

45. Convolution • f(t) and g(t) take on non-zero values for t 0

46. Properties

47. Properties (cont.)

48. Differential eq. • Find f(t)

49. Random Sum

50. Ex.