Introduction to Probability and Stocastic Processes - Part I

Transcription

1 Introduction to Probability and Stocastic Processes - Part I Lecture 1 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark Slides originally by: Line Ørtoft Endelt Introduction to Probability and Stocastic Processes - Part I p. 1/50

2 Set Definitions I A set is a collection of elements. Sets are denoted by CAPITAL letters, and elements by small letters. The symbol means is an element of, and / means is not an element of. The symbol denotes the empty set. The entire space is denoted by S (Sample Space). A set is countable if it is finite, or its elements has a one-to-one correspondence with the integers. A set A is contained in a set B, denoted A B ( or B A), if every element of A is also in B. The following three statements are always satisfied: A S, A and A A Introduction to Probability and Stocastic Processes - Part I p. 2/50

3 Set Definitions II A = B if and only if A B and B A. The union of two sets A and B, denote A B, is the set of all elements that belongs to A or B or both. The intersection of two sets A and B, denote A B, is the set of elements that belong to both A and B. Two sets A and B are mutually exclusive if A B = AB =. The complement, A, of a set A relative to a set S, consists of the elements of S, which are not in A. Introduction to Probability and Stocastic Processes - Part I p. 3/50

4 Set Definitions III The Commutative Laws A B = B A A B = B A The Associative Laws (A B) C = A (B C) = A B C (A B) C = A (B C) = A B C The Distributive laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) Introduction to Probability and Stocastic Processes - Part I p. 4/50

5 Set Definitions IV DeMorgan s Laws (A B) = A B (A B) = A B Introduction to Probability and Stocastic Processes - Part I p. 5/50

6 The Sample Space The Sample Space for an experiment is the set of all possible outcomes of the experiment. The Sample Space is denoted S. An event is a subset of S (incl. S). A class S, of sets defined on S is called completely additive if 1. S S 2. If A k S for k = 1, 2, 3,..., then n k=1 A k S for n = 1, 2, 3, If A S, then A S. where A is the complement of A. Introduction to Probability and Stocastic Processes - Part I p. 6/50

7 Probabilities of Random Events Definition: A probability measure is a set function whose domain is a completely additive class S of events defined on the sample space S such that the measure satisfies the following conditions: 1. P(S) = 1 2. P(A) 0 for all A S ( N 3. P k=1 k) A = N k=1 P(A k) if A i A j = for i j, and N may be infinite A random experiment is completely described by a sample space, a probability measure, and a class of sets forming the domain set of the probability measure. The combination of these three items are called a probabilistic model. Introduction to Probability and Stocastic Processes - Part I p. 7/50

8 Probabilities of Random Events Relative Frequency Definition (Probability based on experiment): The number of times the experiment is performed is n, and n A is the number of times, where the outcome belongs to A S. P(A) = lim n n A n Classical Definition: N is the total number of outcomes, and N A is the number of outcomes that belongs to A S. P(A) = N A N Introduction to Probability and Stocastic Processes - Part I p. 8/50

9 Ex. where the classical definition fails Willard H. Longcora drilled die experiment: Thrown a die with drilled pips over one million times, using a new die every 20,000 throw because the die wore down. Upface Total Rel. freq Classical Introduction to Probability and Stocastic Processes - Part I p. 9/50

10 Probabilities of Random Events I The following is a number of Laws which can be shown using the definition of a Probability measure 1. P( ) = 0 2. P(A) 1 3. P(A) = 1 P(A) 4. If A B then P(A) P(B) 5. P(A B) = P(A) + P(B) P(A B) 6. P(A B) P(A) + P(B) Introduction to Probability and Stocastic Processes - Part I p. 10/50

11 Probabilities of Random Events II 7. If A 1 A 2 A n = S and the A i A j = if i j, then P(A) = P(A S) = P(A (A 1 A 2 A n )) = P((A A 1 ) (A A 2 ) (A A n )) = P(A A 1 ) + P(A A 2 ) + + P(A A n ) 8. ( n P i=1 A i ) = P(A 1 ) + P(A 1 A 2 ) + P(A 1 A 2 A 3 ) + n 1 +P(A n i=1 A i ) Introduction to Probability and Stocastic Processes - Part I p. 11/50

12 Joint and Marginal Probability I Consider experiment E 1 having sample space S 1 consisting of outcomes a 1,a 2,...,a n1 and experiment E 2 having sample space S 2 consisting of outcomes b 1,b 2,...,b n2. The joint sample space of the experiments is defined as S = S 1 S 2 = {(a i,b j ) : i = 1, 2,...,n 1,j = 1, 2,...,n 2 } The probability of A i B j is called the joint probability, and is denoted P(A i B j ), often abbrevated P(A i B j ). Introduction to Probability and Stocastic Processes - Part I p. 12/50

13 Joint and Marginal Probability II If the events A 1,...,A n of the experiment E 1 are mutually exclusive and exhaustive, then for the event B j S 2, and S = S 1 S 2 : P(B j ) = P(B j S) = P(B j (A 1 A 2 A n )) n = P(A i B j ) i=1 Since B j is associated with subexperiment E 2, P(B j ) is called a marginal probability. Introduction to Probability and Stocastic Processes - Part I p. 13/50

14 Conditional Probability Using the following definitions of probability P(AB) = N AB N, P(A) = N A N the probability that B will happen given that A has happened is P(B A) = N AB = N AB/N N A N A /N and the conditional probability of B given A is defined as P(B A) = P(AB) P(A) Introduction to Probability and Stocastic Processes - Part I p. 14/50

15 Joint, Marginal and Conditional Prop. Relationships: 1. P(AB) = P(A B)P(B) = P(B A)P(A) 2. If AB =, then P(A B C) = P(A C) + P(B C) 3. P(ABC) = P(A)P(B A)P(C AB) 4. If B 1,B 2,...,B m are mutually exclusive and exhaustive, then m P(A) = P(A B j )P(B j ) j=1 5. Bayes rule P(B j A) = P(A B j )P(B j ) m j=1 P(A B j)p(b j ) = P(A B j)p(b j ) P(A) Introduction to Probability and Stocastic Processes - Part I p. 15/50

16 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals Introduction to Probability and Stocastic Processes - Part I p. 16/50

17 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals a) Probability of being from manufacturer M 2 and having the defect B 1? P(M 2 B 1 ) =? Introduction to Probability and Stocastic Processes - Part I p. 16/50

18 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals a) Probability of being from manufacturer M 2 and having the defect B 1? P(M 2 B 1 ) = % Introduction to Probability and Stocastic Processes - Part I p. 16/50

19 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals b) Probability of having the defect B 2? P(B 2 ) =? Introduction to Probability and Stocastic Processes - Part I p. 16/50

20 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals b) Probability of having the defect B 2? P(B 2 ) = % Introduction to Probability and Stocastic Processes - Part I p. 16/50

21 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals c) Probability of being from manufacturer M 1? P(M 1 ) =? Introduction to Probability and Stocastic Processes - Part I p. 16/50

22 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals c) Probability of being from manufacturer M 1? P(M 1 ) = % Introduction to Probability and Stocastic Processes - Part I p. 16/50

23 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals d) Probability of having defect B 2 given it is from M 2? P(B 2 M 2 ) =? Introduction to Probability and Stocastic Processes - Part I p. 16/50

24 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals d) Probability of having defect B 2 given it is from M 2? P(B 2 M 2 ) = , 3% Introduction to Probability and Stocastic Processes - Part I p. 16/50

25 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals d) Probability of having defect B 2 given it is from M 2? P(B 2 M 2 ) = P(B 2M 2 ) P(M 2 ) = = Introduction to Probability and Stocastic Processes - Part I p. 16/50

26 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals e) Probability of being from M 1, given it has defect B 2? P(M 1 B 2 ) =? Introduction to Probability and Stocastic Processes - Part I p. 16/50

27 Ex. on joint and marginal probabilities A number of components from manufacturer M i for is grouped in the following classes of defects B j, i,j = {1,...,4}. B 1 B 2 B 3 B 4 B 5 Totals M M M M Totals e) Probability of being from M 1, given it has defect B 2? P(M 1 B 2 ) = % Introduction to Probability and Stocastic Processes - Part I p. 16/50

28 Binary Communication Channel One and zero is transmitted. If A = one transmitted B = one is received P(A) = 0.6 P(B A) = 0.90 P(B A) = 0.05 Introduction to Probability and Stocastic Processes - Part I p. 17/50

29 Binary Communication Channel One and zero is transmitted. If A = one transmitted B = one is received P(A) = 0.6 P(B A) = 0.90 P(B A) = 0.05 Probability a one is received? P(B) = P(B A)P(A) + P(B A)P(A) = = 0.56 Introduction to Probability and Stocastic Processes - Part I p. 17/50

30 Binary Communication Channel One and zero is transmitted. If A = one transmitted B = one is received P(A) = 0.6 P(B A) = 0.90 P(B A) = 0.05 Probability a one was transmitted given a one was received? P(A B) = P(B A)P(A) P(B) = = % Introduction to Probability and Stocastic Processes - Part I p. 17/50

31 Statistical Independence Two events A i and B j are statistically independent if P(A i B j ) = P(A i )P(B j ) P(A i B j ) = P(A i ) Statistical independence is not the same as mutually exclusive!! Example: Tossing a die, let A = {2, 4, 6} and B = {5, 6} Then P(AB) = 1 6 = P(A)P(B) and P(A B) = 1 2 = P(A) A and B are statistically independent but not mutually exclusive. Introduction to Probability and Stocastic Processes - Part I p. 18/50

32 Random Variables Definition:A random variable X is a function X(λ) : S R, such that 1. The set {λ : X(λ) x} is an event for every x R. 2. P(X = ) = P(X = ) = 0. Hence for every A S there corresponds a set T R called the image of A. For every set T R there exists a set X 1 (T) S, called the inverse image of T, which satisfies X 1 (T) = {λ S : X(λ) T } Notation: P(X = x) = P {λ : X(λ) = x}. Introduction to Probability and Stocastic Processes - Part I p. 19/50

33 From Sample Space to Random Variable Tossing a die the Sample Space is: X: The random variable X maps # of eyes on the Up face side of the die to R. Assume the die is fair, then P(X = x i ) = 1 6. Introduction to Probability and Stocastic Processes - Part I p. 20/50

34 Distribution Function Definition:The distribution function of the random variable X is given by F X (x) = P(X x). A distribution function has the following properties: 1. F X ( ) = 0 2. F X ( ) = 1 3. lim ǫ 0 ǫ>0 F X (x + ǫ) = F X (x) 4. F X (x 1 ) F X (x 2 ) if x 1 < x 2 5. P(x 1 < X x 2 ) = F X (x 2 ) F X (x 1 ) Introduction to Probability and Stocastic Processes - Part I p. 21/50

35 Distribution Function for a fair die x i F X (x i ) Introduction to Probability and Stocastic Processes - Part I p. 22/50

36 Joint Distribution Function Definition:The joint distribution function for the two random variables X and Y is given by F X,Y (x,y) = P[(X x) (Y y)] From this definition note, that F X,Y (, ) = 0, F X,Y (,y) = 0, F X,Y (,y) = F Y (y) F X,Y (x, ) = 0, F X,Y (, ) = 1, F X,Y (x, ) = F X (x) Introduction to Probability and Stocastic Processes - Part I p. 23/50

37 Discrete Random Variables Definition:A discrete random variable only takes on a finite set of values. The probability P(X = x i ) for i = 1, 2,...,n is called the probability mass function. A probability mass function has the following properties: 1. P(X = x i ) > 0 for i = 1, 2,...,n n 2. P(X = x i ) = 1 i=1 3. P(X x) = F X (x) = x i x P(X = x i ) 4. P(X = x i ) = lim ǫ 0 ǫ>0 (F X (x i ) F X (x i ǫ)) Introduction to Probability and Stocastic Processes - Part I p. 24/50

38 Probability Mass function for a die P(X = x i ) Fair die x i P(X = x i ) Drilled die x i Introduction to Probability and Stocastic Processes - Part I p. 25/50

39 Relationships for two Random Variables 1. P(X x,y y) = P(X = x i,y = y j ) x i x y j y m 2. P(X = x i ) = P(X = x i,y = y j ) j=1 m = P(X = x i Y = y j )P(Y = y j ) j=1 3. P(X = x i Y = y j ) = P(X = x i,y = y j ) P(Y = y j ) P(Y = y j X = x i )P(X = x i ) = n i=1 P(Y = y j X = x i )P(X = x i ) 4. X and Y are statistically independent if for all i,j P(X = x i,y = y j ) = P(X = x i )P(Y = y j ) Introduction to Probability and Stocastic Processes - Part I p. 26/50

40 Example: Joint distribution Joint distribution for two fair dies: X : # of eyes on the Up Face side of die 1. Y : # of eyes on the Up Face side of die 2. The joint probability mass function for X and Y is P(X = i,y = j) = 1 36 for i,j {1, 2,..., 6} and the joint distribution function F X,Y (x,y) = x i=1 y i= = xy 36 for x,y {1, 2,..., 6} Introduction to Probability and Stocastic Processes - Part I p. 27/50

41 Expected values, mean and variance The average or expected value of a function g of a discrete random variable X is n E{g(X)} = g(x i )P(X = x i ) i=1 The mean is defined as the expected value of the variable: n E{X} = µ X = x i P(X = x i ) i=1 The variance of a discrete random variable is defined as n E{(X µ X ) 2 } = σx 2 = (x i µ X ) 2 P(X = x i ) i=1 σ X is called the standard deviation. Introduction to Probability and Stocastic Processes - Part I p. 28/50

42 Die example!! X : # of eyes on the Up Face side of a fair die. The mean value: E{X} = µ X = 6 x i P(X = x i ) = ( ) 1 6 = 3.5 i=1 The variance E{(X µ X ) 2 } = σ 2 X = 6 i=1 (x i µ X ) 2 P(X = x i ) = If g(x i ) = x 2 i then: E{g(X)} = 6 x 2 i P(X = x i ) = i=1 Introduction to Probability and Stocastic Processes - Part I p. 29/50

43 Expected values, mean and variance I If the probability mass function is not known, but the mean and the varians are known, Tchebycheff s inequality can be used to evaluate the probability of a random variable P[ X µ X > k] σ2 X k The expected value of a function of two random variables is defined as E{g(X,Y )} = n m g(x i,y j )P(X = x i,y = y j ) i=1 j=1 Introduction to Probability and Stocastic Processes - Part I p. 30/50

44 Expected values, mean and variance II The correlation coefficient is defined as ρ XY = E{(X µ X)(Y µ Y )} σ X σ Y = σ XY σ X σ Y σ XY is called the covariance. The correlation coefficient ρ XY [ 1, 1], if X and Y are independent, then ρ XY = 0 and if they are linearly dependent, then ρ XY = 1. Note: ρ XY = 0 does NOT imply statistical independence! Two Random variables are said to be orthogonal if E XY = 0 Introduction to Probability and Stocastic Processes - Part I p. 31/50

45 Expected values, mean and variance III The conditional expected values are defined as E{g(X,Y ) Y = y j } = E{g(X,Y ) X = x i } = it can be shown that n i=1 m j=1 g(x i,y j )P(X = x i Y = y j ) g(x i,y j )P(Y = y j X = x i ) E{g(X,Y )} = E XY {g(x,y )} = E X {E Y X [g(x,y ) X]} The conditional mean value is E{X Y = y j } = µ X Y =yj = i x i P(X = x i Y = y j ) Introduction to Probability and Stocastic Processes - Part I p. 32/50

46 The Uniform Probability Mass Function The Uniform Probability Mass Function is given by Example: Fair die P(X = x i ) P(X = x i ) = 1, i = 1, 2,...,n n x i Introduction to Probability and Stocastic Processes - Part I p. 33/50

47 The Binomial Probability Mass Function The Binomial Probability Mass Function. If P(A) = p, and the experiment is repeated n times, let X be a random variable representing the number of times A occurs, then ( ) n P(X = k) = p k (1 p) n k, k = 1, 2,...,n k where ( n) k = n! k!(n k)!. The mean value and variance of a binomial random variable are µ X = np and σ 2 X = np(1 p) Introduction to Probability and Stocastic Processes - Part I p. 34/50

48 Example: The Binomial Prob. Mass Fct For n = 10 and p = Introduction to Probability and Stocastic Processes - Part I p. 35/50

49 Example: The Binomial Prob. Mass Fct For n = 10 and p = Introduction to Probability and Stocastic Processes - Part I p. 36/50

50 Poisson Probability Mass Function Assume 1. The number of events occurring in a small time interval t λ t as t The number of events occurring in non overlapping time intervals are independent. Then the number of events in a time interval T have a Poisson Probability Mass Function of the form where λ = λ T. P(X = k) = λk k! e λ, k = 0, 1, 2,... The mean and the variance are µ X = σ 2 X = λ. Introduction to Probability and Stocastic Processes - Part I p. 37/50

51 Example: Poisson Prob. Mass Fct For λ = Introduction to Probability and Stocastic Processes - Part I p. 38/50

52 Binary communication system Ia X: input, 0 or 1, with P(X = 0) = 3 4 and P(X = 1) = 1 4 Y : output, due to noise: P(Y = 1 X = 1) = 3 4 and P(Y = 0 X = 0) = 7 8 Find P(Y = 1) and P(Y = 0) P(Y = 1) = P(Y = 1 X = 0)P(X = 0) +P(Y = 1 X = 1)P(X = 1) ( = 1 7 )( ) ( )( ) = P(Y = 0) = 1 P(Y = 1) = Introduction to Probability and Stocastic Processes - Part I p. 39/50

53 Binary communication system Ib X: input, 0 or 1, with P(X = 0) = 3 4 and P(X = 1) = 1 4 Y : output, due to noise: P(Y = 1 X = 1) = 3 4 and P(Y = 0 X = 0) = 7 8 Find P(X = 1 Y = 1) P(X = 1 Y = 1) = = P(Y = 1 X = 1)P(X = 1) P(Y = 1) ( 34 )( 14 ) 9 32 = 2 3 Introduction to Probability and Stocastic Processes - Part I p. 40/50

54 Binary communication system IIa Binary data are sent in blocks of 16 digits over a noisy communication channel. p = 0.1 is the probability that a digit is in error (independent of whether other digits are in error). X: Number of errors per block. X has a binomial distribution. ( ) 16 P(X = k) = (0.1) k (0.9) 16 k, k = 1, 2,...,16 k Find the average number of errors per block E{X} = np = (16)(0.1) = 1.6 Introduction to Probability and Stocastic Processes - Part I p. 41/50

55 Binary communication system IIb X: Number of errors per block. X has a binomial distribution. ( ) 16 P(X = k) = (0.1) k (0.9) 16 k, k = 1, 2,...,16 k Find the variance of X σ 2 X = np(1 p) = (16)(0.1)(0.9) = 1.44 Find P(X 5) P(X 5) = 1 P(X 4) 4 ( ) 16 = 1 (0.1) k (0.9) 16 k k k=0 Introduction to Probability and Stocastic Processes - Part I p. 42/50

56 Continuous Random Variables A continuous random variable can take any value in an interval of the real line. For a continuous random variable the probability density function (pdf) is defined by Properties: 1. f X (x) 0 2. f X(x)dx = 1 f X (x) = df X(x) dx 3. P(X a) = F X (a) = a f X(x)dx 4. P(a X b) = a b f X(x)dx 5. P(X = a) = a a f X(x)dx = lim x 0 f X(a) x = 0 Introduction to Probability and Stocastic Processes - Part I p. 43/50

57 Two Continuous Random Variables The joint probability density function f X,Y (x,y) = d2 F X,Y (x,y) dxdy 0 The joint distribution function can be found as F X,Y (x,y) = y x f X,Y (µ,η)dµdη Since F X,Y (, ) = 1, then f X,Y (µ,η)dµdη = 1 Introduction to Probability and Stocastic Processes - Part I p. 44/50

58 Two Continuous Random Variables I The marginal and conditional density functions are obtained as follows: f X (x) = f Y (y) = f X,Y (x,y)dy f X,Y (x,y)dx f X Y (x y) = f X,Y (x,y), f Y (y) > 0 f Y (y) f Y X (x y) = f X,Y (x,y) f Y (y) = f X Y (x y)f Y (y) f X Y (x λ)f Y (λ)dλ Introduction to Probability and Stocastic Processes - Part I p. 45/50

59 Two Continuous Random Variables II Two random variables are statistically independent if f X,Y (x,y) = f X (x)f Y (y) Introduction to Probability and Stocastic Processes - Part I p. 46/50

60 Expected values I E{g(X,Y )} = µ X = E{X} = g(x,y)f X,Y (x,y)dxdy σ 2 X = E{(X µ X) 2 } = xf X (x)dx σ XY = E{(X µ X )(Y µ Y )} = ρ XY = E{(X µ X)(Y µ Y )} σ X σ Y (x µ X ) 2 f X (x)dx (x µ X )(y µ Y )f X,Y (x,y)dxdy Introduction to Probability and Stocastic Processes - Part I p. 47/50

61 Expected values II Conditional expected values are defined as E{g(X,Y ) Y = y} = g(x,y)f X Y (x y)dx If X and Y are independent then E{g(X)h(Y )} = E{g(X)}E{h(Y )} Introduction to Probability and Stocastic Processes - Part I p. 48/50

62 Example The joint density function of X and Y is Find a: f X,Y (x,y) = axy 1 x 3, 2 y 4 f X,Y (x,y) = 0 elsewhere 1 = = a axy dx dy = a 4y dy = 24a 4 2 y [ x 2] 3 2 dy 1 so a = 1 24 Introduction to Probability and Stocastic Processes - Part I p. 49/50

63 Example (continued) The marginal pdf of X: f X (x) = 24 1 f X (x) = xy dy = x 4 1 x 3 elsewhere The distribution function of Y is F Y (y) = 0 y 2 F Y (y) = 1 y > 4 F Y (y) = 24 1 y xv dx dv = 1 y 6 2 v dv = 12 1 (y2 4) 2 y 4 Introduction to Probability and Stocastic Processes - Part I p. 50/50