Marquette University MATH 1700 Class 9 Copyright 2017 by D.B. Rowe

Transcription

1 Class 9 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 207 by D.B. Rowe

2 Ageda: Recap Chapter Lecture 5.3 cotiued Review Chapter for Eam 2 2

3 Recap Chapter

4 5: Probability Distributios (Discrete Variables) 5.2 Probability Distributios of a Discrete Radom Variable Radom Variables: assumes a uique value for each of the outcomes i the sample space. Probability Fuctio: A rule P() that assigs probabilities to the values of the radom variable. Eample: Let = # of heads whe we flip a coi twice. ={0,,2} 2! P ( )!(2 )! P( )

5 5: Probability Distributios (Discrete Variables) 5.2 Mea ad Variace of a Discrete Radom Variable [ i P( i )] P( ) 2P( 2)... P( ) i For the # of H whe we flip a coi twice discrete distributio: μ = ( ) P( ) + ( 2 ) P( 2 ) + ( 3 ) P( 3 ) μ = (0) P(0) + () P() + (2) P(2) μ = (0) (/4) + () (/2) + (2) (/4) μ = 0 + /2 + /2 μ = P( ) P ( ) P ( 2) P ( 3) 5

6 5: Probability Distributios (Discrete Variables) 5.2 Mea ad Variace of a Discrete Radom Variable i P i P 2 P 2 P i [( ) ( )] ( ) ( ) ( ) ( )... ( ) ( ) For the # of H whe we flip a coi twice discrete distributio: σ 2 = ( -μ) 2 P( ) + ( 2 -μ) 2 P( 2 ) + ( 3 -μ) 2 μ = P( 3 ) σ 2 = (0-) 2 P(0) + (-) 2 P() + (2-) 2 P(2) σ 2 = (-) 2 (/4) + (0) 2 (/2) + () 2 (/4) σ 2 = / /4 σ 2 = /2 σ P( ) P ( ) P ( 2) P ( 3) 6

7 5: Probability Distributios (Discrete Variables) 5.2 Mea ad Variace of a Discrete Radom Variable [ i P( i )] [ P( ) 2 P( 2)... P( )] i Alterate Formula For the # of H whe we flip a coi twice discrete distributio: σ 2 = 2 P( ) + 22 P( 2 ) + 32 P( 3 )-μ 2 μ = σ 2 = [0 2 P(0) + 2 P() P(2)]- 2 σ 2 =[0 (/4) + 2 (/2) (/4)]- σ 2 = 0 + /2 + 4/4 - σ 2 = /2 σ P( ) P ( ) P ( 2) P ( 3) 7

8 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio A eperimet with oly two outcomes is called a Biomial ep. Call oe outcome Success ad the other Failure. Each performace of ept. is called a trial ad are idepedet. Prob of eactly successes! P( ) p ( p)!( )! = umber of trials or times we repeat the eperimet. = the umber of successes out of trials. p = the probability of success o a idividual trial. um( successes) P( successes ad - failures) Bi meas two like bicycle 0,..., (5.5)!!( )! 8

9 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio Flip coi te times.! P( ) p ( p)!( )! =0, =7, p= ! P(7) 7!3! ! P(7) 7!(0 7)! = umber of trials = the umber of successes p = the probability of success P(7) P(7) 024 9

10 Lecture Chapter 5.3 cotiued 0

11 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio Eample: =0, p=0.5 P ( ) These Biomial probabilities ca also be foud i the back of the book. Table 2 i Appedi B Please tur to page 73! P( ) p ( p)!( )!

12 umber of success i trials page 73 prob of success o a trial umber of trials.!!( )! Figure from Johso & Kuby,

13 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio page 73 page 659!! P( P )( ) pp ( p )!(!( )! =2, p=/2 P( ) 0 / 4 2 / 4 2 / 4 2! P(0) / 2 0!(2 0)! ( / 2) 2! P() / 2!(2 )! ( / 2) 2! P(2) / 2 2!(2 2)! ( / 2) Figure from Johso & Kuby,

14 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio page 660 page 73 =0, =7, p=/2! P( ) p ( p)!( )! Figure from Johso & Kuby, 202. P ( )

15 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio! P( ) p ( p)!( )! =2, p=2/3 P( ) 0 / 9 4 / / 9 p=2/ Not every p is o the table. Figure from Johso & Kuby,

16 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio Eample: =0, p=/2 What is the probability of gettig 4, 5, or 6 heads? P(4 6)=P(4)+P(5)+P(6) P(4 6)= P(4 6)= P ( )

17 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio Eample: =0, p=/2 What is the probability of gettig 7 or fewer heads? P( 7)=P(0)+P()+P(2)+P(3)+P(4)+P(5)+P(6)+P(7) P ( )

18 5: Probability Distributios (Discrete Variables) 5.3 The Biomial Probability Distributio Eample: =0, p=/2 What is the probability of gettig 7 or fewer heads? P( 7)=-P( 8) P( 7)= P ( )

19 5: Probability Distributios (Discrete Variables) 5.3 Mea ad Stadard Deviatio of the Biomial Distributio A eperimet with two outcomes does ot have to be H ad T. More geeral tha H ad T, call oe Success ad other Failure. Geerally call the oe we re iterested i the Success. Now that we ve established that the formula works. We ca determie the theoretical mea umber of heads, μ, ad the theoretical variace for the umber of heads, σ 2. 22

20 5: Probability Distributios (Discrete Variables) 5.3 Mea ad Stadard Deviatio of the Biomial Distributio The geeral formula for determiig the theoretical mea μ of a discrete distributio is: [ ip( i)] Ad upo isertio of the Biomial distributio i 0 p! p ( p)!( )! =0, 2 =, 3 =2, (5. (5.7) 23

21 5: Probability Distributios (Discrete Variables) 5.3 Mea ad Stadard Deviatio of the Biomial Distributio The geeral formula for determiig the theoretical variace σ 2 of a discrete distributio is: ( ) P( ) 2 2 Ad upo isertio of the Biomial distributio! ( ) p ( p)!( )! p( p) =0, 2 =, 3 =2, p( p) (5.8) 24

22 5: Probability Distributios (Discrete Variables) 5.3 Mea ad Stadard Deviatio of the Biomial Distributio Eample: Before, usig, we foud. Now usig, we get. Before, usig 2 / 2 foud. [ P( )] 0 p (2) (/ 2) Now usig, p( p) [( ) P( )] 2 (2) (/ 2) (/ 2) / 2, we we get. 0 2 P( ) =2 = p=/2 25

23 5: Probability Distributios (Discrete Variables) 5.3 Mea ad Stadard Deviatio of the Biomial Distributio Eample: =0, p=/2 P ( ) What are μ ad σ 2? 2 p p( p) 26

24 5: Probability Distributios (Discrete Variables) Questios? Homework: Chapter 5 # 5, 7, 9, 29, 3, 43 55a,b, 63, 77, 85, 89 Read Chapter

25 Review Chapters (Eam 2 Chapters) Just the highlights! 29

26 3: Descriptive Aalysis ad Bivariate Data 3.2 Liear Correlatio Liear Correlatio, r, is a measure of the stregth of a liear relatioship betwee two variables ad y. r Will discuss its computatio i a miute. r 0 r 0.5 r 0.8 r 0.5 r 0.8 Figure from Johso & Kuby,

27 3: Descriptive Aalysis ad Bivariate Data 3.2 Liear Correlatio Eample: Figure from Johso & Kuby,

28 3: Descriptive Aalysis ad Bivariate Data 3.2 Liear Correlatio Eample: (35) ( ) i i i i 0 SS (380) ( ) i i i i 0 SS y y y (35)(380) SS( y) i yi i yi i i i 0 SS( y) 99.0 r 0.84 SS( ) SS( y) (396.9)(858.0) i i i i i y y i 2 i i 2 i y i i 4257 Figure from Johso & Kuby,

29 3: Descriptive Aalysis ad Bivariate Data 3.4 Liear Regressio y Marquette Uiversity MATH 700 is estimated y-itercept is estimated slope. ŷ b b We try differet lies util we fid the best oe,. 3 2 y b 0 b 0 Move lie util sum of the squared residuals is a miimum. i 2 i 33

30 3: Descriptive Aalysis ad Bivariate Data 3.4 Liear Regressio (,y) pairs: (,),(3,2),(2,3),(4,4) Plotted poits. The lie goes through (, y). SS( y) i yi i yi i i i 2 SS( ) i i i i 2 The is b =0.8. The y - itercept b 0 =0.5. Two poits (2.5,2.5) ad (0,.5). ru rise SS( y) b SS( ) b y b 0 34

31 3: Descriptive Aalysis ad Bivariate Data 3.3 Liear Regressio Eample: Our data! Height vs. Weight b TuTh 02 SS( y) SS( ) 04.7 (, y) uits of lbs/i y b b0 (50.8) (2.7)(68.) 3.5 poit-slope formula 35

32 3: Descriptive Aalysis ad Bivariate Data Questios? Homework: Chapter 3 # 33, 44, 53, 59, 75 Read 4. ad

33 4: Probability 4. Probability of Evets A eperimet is a process by which a measuremet is take or observatios is made. i.e. flip coi or roll die A outcome is the result of a eperimet. i.e. Heads, or 3 Sample space is a listig of possible outcomes. i.e. S={H,T} A evet is a outcome or a combiatio of outcomes. i.e. eve umber whe rollig a die 37

34 4: Probability 4. Probability of Evets Property : Property 2: 0 PA ( i ) PO ( ) i Approaches to probability. () Empirical (AKA eperimetal) empirical probability of i A A i are evets,..., umber of times A occured umber of trials (2) Theoretical (AKA classical or equally likely) umber of times A occus i sample space theoretical probability of A umber of elemets i the sample space i O i are outcomes 38

35 4: Probability - Empirical 4. Probability of Evets Law of large umbers Had computer flip a sigle coi 000 times. P'( H) # of heads # coi flips Flip # o ais P'(H) o y ais. This shows covergece to true value of /2. bad i short ru good i log ru 39

36 4: Probability 4. Probability of Evets So let's flip a coi three times. Ca flip three times. H T H T H T H T H T H T H T S Sample space: listig of outcomes for 3 flips { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} P( HHH ) # times HHH occurs i # elemets i S S 40

37 4: Probability 4.3 Rules of Probability Eample: Pick Card, A=Heart, B=Ace P( A) P( A) P( A or B) P( A) P( B) P( A ad B) P( A ad B) P( A) P( B A) P( A B) P( A ad B) P( B) Figure from Johso & Kuby,

38 4: Probability 4.4 Mutually Eclusive Evets Mutually eclusive evets: Evets that share o commo elemets I algebra: P( A ad B) 0 (o overlap) I words:. If oe evet has occurred, the other caot. 2. Noe of the elemets i oe is i other. 3. I Ve diagrams, o itersectio. 4. Itersectio of evets has a probability of zero. 42

39 4: Probability 4.5 Idepedet Evets Idepedet evets: the occurrece or ooccurece of oe gives o iformatio about occurrece for the other. P( A) P( A B) P( A ot B) Depedet evets: occurrece of oe evet does have a effect o the probability of occurrece of the other evet. P( A) P( A B) Special multiplicatio rule: Let A ad B idepedet evets i a sample space S. P( A ad B) P( A) P( B) 43

40 4: Probability Questios? Homework: Chapter 4 # 59, 63, 65, 69, 85, 89, 9, 97, 05, 07, 3 44