Propagation of error for multivariable function

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1 Propagaton o error or multvarable uncton ow consder a multvarable uncton (u, v, w, ). I measurements o u, v, w,. All have uncertant u, v, w,., how wll ths aect the uncertant o the uncton? L tet) o (Equaton (.48) w w v w u u ± L w w v w u u ),... w, v, (u Ths works or cases lke sstematc errors, when the errors o most o the varables have the same sgn. For cases lke random errors, ths overestmate and gve an upper bound o the actual error: bound o the actual error: W ll t d th d l t th w w v w u u L We wll stud the case o random error later n the course.

2 Eample I, calculate n terms o and. ) ( - ) ( - Add not subtract Add, not subtract

3 Eample I z, calculate the racton error n. (z) (z) (z) z z z z z z z (z) (z) (z) z z z z z z For product o varables the resultant racton error s the sum o the racton For product o varables, the resultant racton error s the sum o the racton errors o the varables.

4 Eample (Problem.7(d) o tet) A student t makes the ollowng measurement: a 5 ± cm, b 8 ± cm, c ± cm, t.0 ± 0.5 s, m 8 ± gram Compute the quantt mb/t wth ts uncertantes and percentage uncertantes mb t m m b b t b m mb m b t t t t t cm - g/s cm - g/s 40cm - g/s mb cm - g/s 0 cm - g/s t 0 ± 40 cm - g/s Percentage uncertant % % 40%

5 Topcs Part Sngle measurement. Basc stu (Chapter and ). Propagaton o uncertantes (Chapter ) Part Multple measurements as ndependent results. Mean and standard devaton (Chapter 4). Basc on probablt dstrbuton uncton (not n tet eplctl). The Bnomal dstrbuton (Chapter 0) 4. The Posson dstrbuton (Chapter ) 5. ormal dstrbuton (rst hal o Chapter 5). χ test how well does the data t the dstrbuton model? (Chapter ) Part Multple measurements as one sample. Central lmt theorem (not n tet eplctl). ormal dstrbuton (second hal o Chapter 5). Propagaton o error (Chapter ) 4. Rejecton o data (Chapter ) 5. Mergng two sets o data together (Chapter 7) Part 4 Dependent varables. Curve ttng (Chapter 8). Covarance and correlaton (Chapter 9)

6 Mean and standard devaton The mean and varance o a collecton o data o sample sze are dened as: σ σ mean o varance o ( ( sample dstrbuton - ) - ) the measurement s called the standard devaton

7 Eample Consder a collecton o data {4.0, 4.5,.9, 4., 4.,.7} Mean Standard devaton (4.0-4.) ( ) (4.5-4.) 0.07 (4.- 4.) (.9-4.) (.7-4.) We epect the net measurement wll probabl all n the range o 4.± 0.

8 Two Dentons o Standard Devaton σ ( - ) s called the populaton standard devaton σ ( - - ) s called the sample standard devaton For large, populaton standard devaton sample standard devaton. At ths pont, we wll use populaton standard devaton, but later sample standard devaton wll become more mportant.

9 Populaton Standard Devaton ( - ) Populaton standard devaton σ s a measure o the actual spread o the data. When the sample sze s, the populaton standard devaton equals to 0, because there s no spread n the data. However, ths does not mean ths s a good collecton o data.

10 Hstogram Frequenc bn Varable Assumngallbns have the samesze, sze, ou do one more measurement, probabltor ts value to all n bn P. Ths s probabl the best ou can guess rom the j data. j

11 ormalzed Hstogram Varable Assumng all bns have the same sze, ou can just read P rom the vertcal as o a normalzed hstogram. P ote that P j j

12 Probablt Denst Functon P() When, we can make the bn sze 0 (d). When ths happens, the normalzed hstogram wll become a probablt denst uncton: P P()d j Pj P()d Random varable Unortunatel, we wll never know P() because t s mpossble to sats.

13 Eample For a ar dce, the probablt dstrbuton uncton s dscrete or ntegers rom to : P() / 4 5

14 Eample Consder probablt denst uncton: P() 0 Probablt or to occur between 0.5 and

15 ormalzaton condton o the probablt denst uncton I [, b ]. Dscrete case : a b p( Contnuous case : b a a ) p() d

16 Eample p For a ar dce, the probablt dstrbuton uncton s dscrete or ntegers rom to : P() / 4 5 ) ( p

17 Eample Consder probablt denst uncton: P() 0 d ) ( d d P()

18 Eample A ar dce s thrown, the out come s, I pa ou $5.9. Otherwse, ou pa me $.00. What s the epectaton o our earnng? {,,,4,5,}. I () our earnng,. () 5.9, ()()(4)(5)().0 Snce the dce s ar, p()p()p()p(4)p(5)p()/ <()>$5.9/ $.0/ ) $ $.0/ $.0/ $.0/ $.0/ $5.9/ $5.0/ $0.5 I epect ou to earn $0.5 per game.

19 Mean The mean o a dstrbuton p() s just the epectaton o, <> or. I [ a, b ]. Dscrete case : < b > Contnuous case : a p( ) < > b p() d a

20 Eample For a ar dce, the probablt dstrbuton uncton s dscrete or ntegers rom to : P() () / 4 5 Mean < > p( ) The mean s the average ace value ater man trals.

21 Eample Consder probablt denst uncton: P() 0 d ) ( d d () d ) ( d d P() 0 0 > <

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