Probability Theory and Statistics

Transcription

1 robability Theory and Statistics Kasper K. erthelsen, Dept. For Mathematical Sciences Literature: Walpole, Myers, Myers & Ye: robability and Statistics for Engineers and Scientists, rentice Hall, 8th ed. Slides and lecture overview: Lecture format: 2x45 min lecturing followed by exercises in group rooms 1 Lecture1

2 STTISTICS What is it good for? Forecasting: Expectations for the future? How will the stock markets behave?? nalysis of sales: How much do we sell, and when? Should we change or sales strategy? C Quality control: What is my rate of defective products? How can I best manage my production? What is the best way to sample? Lecture1

3 robability theory Sample space and events Consider an experiment Sample space S: S Event : S VENN DIGRM Example: S{1,2,,6} rolling a dice S{plat,krone} flipping a coin Example: {1,6} when rolling a dice Complementary event : S Example: {2,3,4,5} rolling a dice 3

4 robability theory Events Example: Rolling a dice S{1,2,3,4,5,6} {2,4,6} {1,2,3} S Intersection: {2} Union: {1,2,3,4,6} Disjoint events: CD Ø C{1,3,5} and D{2,4,6} are disjoint S C D 4

5 robability theory Counting sample points Ways of placing your bets: Guess the results of 13 matches ossible outcomes: 1 X 2 Home win 3 possibilities 3 possibilities Draw way win 3 possibilities nswer: The multiplication rule 5

6 robability theory Counting sample points Ordering n different objects Number of permutations??? There are n ways of selecting the first object n -1 ways of selecting second object 1 way of selecting the last object n factorial n n -1 1 n! ways The multiplication rule 3! 6 6

7 robability theory Counting sample points Multiplication rule: If k independent operations can be performed in n 1, n 2,, n k ways, respectively, then the k operations can be performed in n 1 n 2 n k ways Tree diagram: 7 T H T H T H T H T H T H T H Flipping a coin three times Head/Tail possible outcomes

8 robability theory Counting sample points Number of possible ways of selecting r objects from a set of n destinct elements: Ordered Unordered n Without replacement r n! n r! n n! r r! n r! With replacement r n - 8

9 robability theory Counting sample points Example: nn, arry, Chris, and Dan should from a committee consisting of two persons, i.e. unordered without replacement. Number of possible combinations: 4 4! 2 2!2! Writing it out : C D C D CD 6 9

10 robability theory Counting sample points Example: Select 2 out of 4 different balls ordered and without replacement 4! Number of possible combinations: ! Notice: Order matters! 12 10

11 robability theory robability Let be an event, then we denote the probability for S It always hold that 0 < < 1 Ø 0 S 1 Consider an experiment which has N equally likely outcomes, and let exactly n of these events correspond to the event. Then n N # successful outcomes # possible outcomes Example: Rolling a dice even number

12 12 robability theory robability Example: Quality control batch of 20 units contains 8 defective units. Select 6 units unordered and without replacement. Event : no defective units in our random sample. 20 Number of possible samples: N # possible 6 12 Number of samples without defective units: n !6!14! !6!20! # successful

13 robability theory robability Example: continued Event : exactly 2 defective units in our sample Number of samples with exactly 2 defective units: 12 8 n !8!6!14! !8!2!6!20! 6 # successful 13

14 robability theory Rules for probabilities Intersection: If and are disjoint: Union: + In particular:

15 robability theory Conditional probability Conditional probability for given : where > 0 ayes Rule: Rewriting ayes rule: + 15

16 robability theory Conditional probability Example page 59: The distribution of employed/unemployed amongst men and women in a small town. Employed Unemployed Total Man Woman Total man & employd 460/ man employed 76.7% employd 600/ man & unemployed 40/ man unemployed 13. 3% unemployed 300/

17 robability theory ayes rule Example: Lung disease & Smoking ccording to The merican Lung ssociation 7% of the population suffers from a lung disease, and 90% of these are smokers. mongst people without any lung disease 25.3% are smokers. Events: robabilities: : person has lung disease 0.07 : person is a smoker What is the probability that at smoker suffers from a lung disease?

18 robability theory ayes rule extended version 1,, k are a partitioning of S S 6 Law of total probability: k i 1 i i ayes formel udvidet: 2 18 r k i 1 r i r i

19 robability theory Independence Definition: Two events and are said to be independent if and only if or lternative Definition: Two events and are said to be independent if and only if Notice: Disjoint event mutually exclusive event are dependent! 19

20 robability theory Conditional probability Example: Employed Unemployed Total Man Woman Total / 900 man employed 76. 7% 600/ 900 man 500/ % Conclusion: the two events man and employed are dependent. 20

21 21 robability theory Rules for conditional probabilities C C C C robability of events and happening simultaneously robability of events, and C happening simultaneously roof: C C C C C C k k k k k k General rule: