God doesn t play dice. - Albert Einstein

Transcription

1 ECE 450 Lecture 1 God doesn t play dice. - Albert Einstein As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality. Lecture Overview Announcements Set theory review - Albert Einstein Vocabulary: experiments, outcomes, trials, events, sample space 3 axioms of probability Combinatorics Probability what is it? (4 approaches) EE Application: Information Theory ECE 450 D. van Alphen 1

2 Announcements Regular Office Hr:,, JD 4414 Syllabus Highlights Grading HW Due Dates Recorded Lectures and Tutorials Course Web Page: (Follow links: Current Semester ECE 450) ECE 450 D. van Alphen 2

3 Set Theory On your own time, review set complements, unions, intersections, subsets, set differences, and Venn diagrams from text, pp Recall: Sets A and B are mutually exclusive (m.e., or disjoint) iff: A B = F (the empty set). De Morgan s Laws (A B) = A B (A B) = A B Recall that a set with n elements has subsets. ECE 450 D. van Alphen 3

4 Vocabulary for Probability An experiment is some action that has outcomes (z, zeta) belonging to a fixed set of possible outcomes called the sample space or the universal set or the probability space, S. Each single performance of the experiment is called a. Chance experiment = random experiment, denoted E Before performing the experiment, the actual outcome is unknown; ECE 450 D. van Alphen 4

5 Examples of Experiments Example 1: E 1 = single toss of a die S = { } (sample space) S is finite, countable Example 2: E 2 = turning on radio receiver at time t = 0; measure voltage at certain point in circuit, t seconds later; define the outcome z(t) = v(t), where t is fixed; S = {v: - < v < } (sample space) uncountably infinite (ignoring measurement limits) ECE 450 D. van Alphen 5

6 Examples of Experiments, continued Example 3: E 3 : count the number of photo-electrons, (e), emitted by a particular surface when a particular light beam falls on it for t seconds; define the outcomes z 0 : 0 e's counted, z 1 : 1 e counted, z 2 : 2 e's counted, S = { } countably infinite ECE 450 D. van Alphen 6

7 More Probability Vocabulary Any subset of the sample space is called an. Thus, A is an event if A S. The elements of the event, A, are the individual outcomes, z, belonging to A. An experiment with n possible outcomes has events associated with it. Example 1, cont.' : A = an odd # appears" = { } B = an even # appears" = { } = A' (A-complement) ECE 450 D. van Alphen 7

8 Examples of Events & More Vocabulary Example 2, cont.' : A = voltage between 2 and 4, inclusive = {v: } B = voltage greater than 3" = {v: } Example 3, cont. : A = fewer than 4 e's counted" = B = a negative # of e's counted = F (the null set or empty set) We say event A occurs whenever any outcome in A occurs Elementary events are those that consist of a single outcome; compound events consist of several outcomes. ECE 450 D. van Alphen 8

9 Axioms of Probability Axiomatic approach due to Kolmogorav (a Russian mathematician, early 1900 s) A probability is a # assigned to an event, A, according to three rules or axioms Axiom 1: Pr(A) 0 (No negative probabilities) Axiom 2: Pr(S) = (Something has to happen) Axiom 3: If A & B are m.e., then Pr(A B) = (For 2 m.e. events, probabilities are additive.) We say event A occurs with probability Pr(A) ECE 450 D. van Alphen 9

10 Corollaries to the Axioms Corollary 1: Pr[A'] = 1 - Pr[A] Proof: Pr(S) = Pr[A' A] = Pr(A') + Pr(A) (why? ) 1 = Pr(A') + Pr(A) (why? ) Pr(A') = 1 - Pr(A) Example: Consider a 52-card deck. Pr(ace) = 4/52 = 1/13 (since there are 4 aces in the deck) Pr(not getting an ace) = Pr(2, 3,, 10, J, Q, K) = 1 - = (by cor. 1) Note that the events {ace} and {2,, 10, J, Q, K} are complementary events ECE 450 D. van Alphen 10

11 Corollaries, continued Corollary 2: 0 Pr(A) 1 Proof: Ax. 1; Pr(A) = 1 - Pr(A') (Cor. 1) 0 (Ax. 1) 1 Corollary 3: Pr(F) = 0 Proof: Pr(S) = Pr(S F) = Pr(S) + Pr(F) (since S, F m.e.) ECE 450 D. van Alphen 11

12 Corollaries, continued Corollary 4: Pr(A B) = Pr(A) + Pr(B) - Pr(A B) Proof: Pr(A B) = Pr(A (B A )) = Pr(A) + Pr(B A ) (m.e.) (1) Venn Diagram: S A B (to be completed in class) ECE 450 D. van Alphen 12

13 Corollaries, continued Similarly: Pr(B) = Pr((A B) (A B)) = Pr(A B) + Pr(B A ) Venn Diagram: S A (m.e.) (2) Now subtract equation (1) from equation (2): Pr(B) - Pr(A B) = Pr(A B) - Pr(A) (proving cor. 4) B (to be completed in class) ECE 450 D. van Alphen 13

14 Example (verifying the corollary) Experiment: Toss one die; Find Pr(A B) for A, B below: Let A = {1, 3}, B = {3, 5} Note: A B = {3} Pr(A) = Pr({1} {3}) = Pr{1} + Pr{3} = 1/6 + 1/6 = 1/3 Similarly, Pr(B) = 1/3 Pr{1, 3, 5} = Pr(A B) = Pr(A) + Pr(B) - Pr(A B) = 1/3 + 1/3 - Pr{3} = 1/3 + 1/3-1/6 = 3/6 = ½ (agreeing with our intuition) ECE 450 D. van Alphen 14

15 Combinatorics, Part 1: Combinations (Binomial Coefficients) nc k = "n choose k = n k (n!) (k!) (n k)! = # of ways to choose k objects out of n available objects if the order of the objects doesn t matter = combination of n objects, taken k at a time = # of subsets of size k for a set with n elements Example: # of possible 5-card poker hands: 52C 5 = (MATLAB): >> nchoosek(52,5) = 2,598,960) 52 5 ( ( ) ( ) ) ECE 450 D. van Alphen 15

16 Combinatorics Example: 5-card Poker Example: Pr(3 Spades in 5-card poker hand) = numerator = # ways to choose 3 Spades and 2 non-spades denominator = # of possible 5-card poker hands ECE 450 D. van Alphen 16

17 Combinatorics Example: 5-card Poker Example: Pr(full house) =??? (3 of one rank, 2 of another; e.g. KKK66) 13 # of ways to choose the first rank: = # of ways to choose the second rank: = # of ways to choose 3 of first kind: = # of ways to choose 2 of second kind: = 12 Pr(full house) = 1.44 x 10-3 ranks: numerical values of the cards, as opposed to the suits ECE 450 D. van Alphen 17

18 Combinatorics, Part 2: Permutations or Arrangements np k = n! (n k)! = permutation of n objects taken k at a time = # of ways to arrange k out of n objects, assuming that the order matters Example 1: # of possible license plates if they are formed from 26 letters of the alphabet and are 5 letters in length, and no letter can be repeated 26P 5 = 26!/21! = = 7,893,600 ECE 450 D. van Alphen 18

19 Combinatorics Examples, continued Example 2: # of distinct seating arrangements possible for a group of 6 students, all 6 in a row: 6P 6 = 6! = = 720 Example 3: # of distinct seating arrangements possible for 2 students in a row, chosen from a group of 6 students 6P 2 = 6!/4! = 6 5 = 30 Summary: use combinations when counting the number of ways to select objects if order doesn t matter, as in card games; use permutations when counting the number of ways to arrange objects, when order does matter. ECE 450 D. van Alphen 19

20 Interpretations of Probability: A. Classical Concept The classical concept assumes all outcomes are equally likely # of outcomes in A Pr(A) = # of possible outcomes in S Justified (for some problems) by the Principle of Indifference or Maximum Ignorance : no reason to favor one outcome over another Usually applied to gambling problems: dice, cards, coins, Example: Pr(bridge hand of 13 cards out of 52 has exactly one ace); solution follows ECE 450 D. van Alphen 20

21 Classical Probability: Example Pr(bridge hand of 13 out of 52 cards has 1 ace) = # of bridge hands # of possible with exactly 1ace bridge hands = =.439 ECE 450 D. van Alphen 21

22 Interpretations of Probability: B. Relative Frequency Concept (von Mises) Repeat an experiment N times; suppose (for example) that there are 4 possible outcomes, or elementary events, called A, B, C, and D. Let N A be the # of times event A occurs; similarly define N B, N C, and N D. Clearly, N = N A + N B + N C + N D. Define the relative frequency of event A as: r(a) = N A /N Relative frequency approach: Pr( A) lim r( A) N ECE 450 D. van Alphen 22

23 Relative Frequency Concept, continued Concept: Best predictor of future performance is past performance Relative frequency interpretation justifies Monte-Carlo Experiments (& thus computer simulations) Typical application: actuarial predictions Example: Pr{a 40-yr. old man dies within 1 yr.} = (# of 40-yr. old men who died in calendar year x) (# of 40-yr. old men at start of calendar year x) ECE 450 D. van Alphen 23

24 Interpretations of Probability: C. Distribution Concept Think of 1 unit of sand, representing the probability, to be distributed over sample space S 1 unit of sand S Sand is piled highest over the most likely outcomes in S ECE 450 D. van Alphen 24

25 Interpretations of Probability: D. Measure of Likelihood View Probability is a function whose domain is the sample space and whose range is the set of real numbers between 0 and 1: Impossible events 0 Unlikely events near 0 Very likely events near 1 Certain events 1 ECE 450 D. van Alphen 25

26 EE Application: Information Theory (Subset of CommunicationTheory) Channels can only accommodate so much information ( There exists an information capacity and maximum rate.) How do we measure information? Some concepts: Communication of information prior uncertainty (Ex: whistle the musical note F # ) Prior uncertainty about outcome surprise on occurrence of event e.g., ask: Will I believe in n years? ECE 450 D. van Alphen 26

27 Information Theory Concepts & Definition n =1: yes little surprise or information n = 10: yes a little more information n=100: yes very much surprise or information Thus, less likely events yield greater surprise more information Definition: The information in event A is given by I(A) log 1 Pr(A) log(pr(a)) ECE 450 D. van Alphen 27

28 Information, continued Units of measure for information in event A: I(A) = - Log[Pr(A)] bits if log is base 2 nats if log is base e (natural log) hartleys if log is base 10 (common log) Example 1: Binary Alphabet, S = {0,1} (Think of communicating a string of 1's and 0's, say ASCII, where 1's and 0's are equally likely.) Symbol, s Pr(s) I(s) 0 ½ Log 2 ( ) = 1 bit 1 ½ Log 2 ( ) = 1 bit Average info. per symbol: 1 bit ECE 450 D. van Alphen 28

29 Information, continued Example 2: Binary Alphabet, S = {0,1} This time we'll still send a stream of 1's and 0's, but they are not equally likely; say Pr(0) = ¼, Pr(1) = ¾ Symbol, s Pr(s) I(s) 0 ¼ Log 2 ( ) = bits 1 ¾ Log 2 ( / ) =.42 bits Average info. per symbol: 1/4(2) + 3/4(.42) =.815 bits Recall: To convert logs from one base to another log b (x) = ECE 450 D. van Alphen 29

30 Information & Entropy Definition: The entropy of the source, S, is the average information per symbol, given by H(S) = ss I(s)Pr(s) sym. info. For our examples prob(symbol) Due to bandwidth constraints, a source with a large entropy is desirable. Equally likely symbols H(s) = 1 bit/symbol Pr(0) = ¼, Pr(1) = ¾ H(s) =.815 bits/symbol ECE 450 D. van Alphen 30

31 Review Pr(A B) = Pr(A ) = (general rule) Combination of n things taken k at a time: = Information in the event A, I(A) = Entropy in source S with symbols s: H(S) = ECE 450 D. van Alphen 31