Rounding Answers. => Z=12.03±0.15 cm

Transcription

1 Roudig Aswers The ucertaity should be rouded off to oe or two sigificat figures. If the leadig figure i the ucertaity is 1, we use two sigificat figures, otherwise we use oe sigificat figure. The aswer should be rouded to match the ucertaity. Z= cm, ΔZ=0.153 cm => Z=12.03±0.15 cm 31-Oct-16 Ucertaities, Zheyu Ye 1

2 Operatioal Amplifiers Zheyu Ye The Art of Electroics by Horowitz ad Hill wikipedia. October 10, 2016 Operatioal Amplifiers, Zheyu Ye 2

3 Operatioal Amplifiers Golde Rules: I. The voltage differece betwee the iputs is zero. II. The iputs draw o curret. Rules apply whe the iputs ad outputs are withi the supply voltages. October 10, 2016 Operatioal Amplifiers, Zheyu Ye 3

4 Ivertig Amplifier 31-Oct-16 Statistics-III, Zheyu Ye 4

5 Peak Output vs Frequecy 31-Oct-16 Statistics-III, Zheyu Ye 5

6 Peak Output vs Frequecy 31-Oct-16 Statistics-III, Zheyu Ye 6

7 Itegrator/Differetiator Itegrator V out = 1/(RC) * it V i dt Differetiator V out =(RC) * dv i /dt October 10, 2016 Operatioal Amplifiers, Zheyu Ye 7

8 Itegrator/Differetiator 31-Oct-16 Statistics-III, Zheyu Ye 8

9 Impedace Z=V/I Resistor: Z=R Capacitor: Z=1/(jωC) Iductor: Z=jωL September 12, 2016 Liear Circuits, Zheyu Ye 9

10 Statistics - III Zheyu Ye R.J.Barlow Statistics: a guide to the use of statistical methods i the physical scieces Chapter Oct-16 Statistics-III, Zheyu Ye 10

11 Some Estimators Mea estimator ˆµ x ( ) = x 1 + x x N N = x is cosistet ad ubiased. Variace estimator ˆ V x ( ) = 1 N ( x i µ ) 2 is cosistet ad ubiased. ˆ V ' = 1 N ( x i x) 2 V ˆ '' = 1 ( x i x) 2 N 1 is cosistet but biased. is cosistet ad ubiased. 31-Oct-16 Statistics-III, Zheyu Ye 11

12 Likelihood Fuctio Probability distributio fuctio for radom umber x P x;a ( ) Likelihood fuctio is the probability for observig a particular data sample (x 1, x 2,, x N ) L x 1, x 2,..., x N ;a ( ) = P x 1 ;a ( )P x 2 ;a ( )...P x N ;a ( ) There is a limit to the accuracy of a estimator miimum variace boud (MSB). If a estimator s variace V equals to MSB, the estimator is efficiet. Otherwise, its efficiecy is MVB/V 1 MSB = 1 d 2 l L / da 2 31-Oct-16 Statistics-III, Zheyu Ye 12

13 Maximum Likelihood Method ML fid the parameter values that maximize likelihood fuctio. For a Gaussia probability distributio fuctio f ( x;µ,σ ) = 1 σ 2π exp x µ 2σ 2 ( ) 2 ll = Nlσ + 6 (x ( μ) + ll μ / / (01 = 6 (x ( μ) (01 σ + 2σ + ll σ = N σ 6 (x ( μ) + / μ" #$ = ( x ( N σ" + #$ = / (01 (x ( μ) N σ 9 (01 31-Oct-16 Statistics-III, Zheyu Ye 13 +

14 Maximum Likelihood Method ML fid the parameter values that maximize likelihood fuctio. For a Gaussia probability distributio fuctio f ( x;µ,σ ) = 1 σ 2π exp x µ 2σ 2 ( ) 2 ll = Nlσ + 6 (x ( μ) + / (01 2σ + + ll μ + = N σ + V(μ" #$ ) σ+ N / (01 + ll σ + = N σ (x ( μ) + σ = V(σ" #$ ) σ+ 2N 31-Oct-16 Statistics-III, Zheyu Ye 14

15 Maximum Likelihood Method ML fid the parameter values that maximize likelihood fuctio. For a Gaussia probability distributio fuctio f ( x;µ,σ ) = 1 σ 2π exp x µ 2σ 2 ( ) 2 ll = Nlσ + 6 (x ( μ) + / (01 2σ + At large N, ML is ubiased ad efficiet: V(μ" #$ ) = σ+ N V(σ" #$ ) = σ+ 2N 31-Oct-16 Statistics-III, Zheyu Ye 15

16 Maximum Likelihood Method Taylor expasio: f x = f a + a 1! x a + a 2! (x a) + + ll μ = ll F0FGHI + JKLM$ KF F0FG HI N μ μ" #$ + JKO LM$ KF O F0FGHI N (FJFG HI ) O +! + Whe μ = μ" #$ + V(μ") ll μ = ll F0FGHI + + ll μ + F0FGHI N μ μ" #$ + 2! + = ll F0FGHI Oct-16 Statistics-III, Zheyu Ye 16

17 Least Squares Method LS fid the parameter values that miimize the sum of the squares of the differece betwee a set of measuremets ad their predicted values. / χ + = 6 y ( f(x ( ;a) (01 σ ( + 31-Oct-16 Statistics-III, Zheyu Ye 17

18 Least Square Method A particle moves alog a track with a costat speed. The positios at certai fixed times were recorded as below with a error of 2mm. Use the LS method to determie the speed. Time (s) Distace (m) Oct-16 Statistics-III, Zheyu Ye 18

19 Least Square Method A particle moves alog a track with a costat speed. The positios at certai fixed times were recorded as below with a error of 2mm. Use the LS method to determie the speed. Time t (s) Distace d (mm) d = v N t χ + = (11 υ)+ (19 2υ) (33 3υ)+ (40 4υ) (49 5υ) Oct-16 Statistics-III, Zheyu Ye 19

20 Least Square Method A particle moves alog a track with a costat speed. The positios at certai fixed times were recorded as below with a error of 2mm. Use the LS method to determie the speed. Time t (s) Distace d (mm) Oct-16 Statistics-III, Zheyu Ye 20

21 Maximum Likelihood Method A particle moves alog a track with a costat speed. The positios at certai fixed times were recorded as below with a error of 2mm. Use the LS method to determie the speed. Time (s) Distace (m) d = v N t f d ( ; v N t (, 2 = 1 2 2π exp (d ( v N t ( ) / ll = C + 6 d ( υ N t + ( (01 = C χ+ 31-Oct-16 Statistics-III, Zheyu Ye 21

22 ML vs LS Methods I a liear model, if the errors belog to a ormal distributio the Least Squares estimators are also the Maximum Likelihood estimators Ucertaity of the LS estimators ca be umerically determied by fidig parameter values which give χ + + = χ c(m + 1 (equivalet to ll c(m + 0.5) 31-Oct-16 Statistics-III, Zheyu Ye 22