Statistics Spring MIT Department of Nuclear Engineering

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1 Statstcs.04 Sprng S00

2 Statstcs/Probablty Analyss of eperments Measurement error Measurement process systematc vs. random errors Nose propertes of sgnals and mages quantum lmted mages.04 S00

3 Probablty Dstrbuton Probablty of fndng between and + d or between and dscrete data normalzaton + P d P P P d P d.04 S00

4 .04 S00 Epected Value [ ] [ ] d P E E d P f f E P d P E r 0 0 r th moment mean

5 .04 S00 Varance Second moment about the mean [ ] d P d P d P d P d P E standard devaton

6 Covarance cov, y ρ E [ y ] cov, y y y ρ + ρ mples lnear correlaton If and y are ndependent, then ρ 0, but ρ 0 does not mean that and y are ndependent, only that they are lnearly ndependent. Eample: y for whch ρ 0.04 S00

7 Bnomal Dstrbuton Each term comes from the bnomal epanson [ p + p] N N! r! N r! p r p N r.04 S00

8 Bnomal Probablty Dstrbuton P r r p N! r! N r! number of p r p successes N r probablty of success n sngle tral N number of trals r r rp r r Np P r Np p.04 S00

9 Eamples of Bnomal Dstrbuton.04 S00

10 Posson Dstrbuton as Lmt to Bnomal p N 0 but mean, Np, remans fnte P r r e r! andfor n measurements n.04 S00

11 Eamples of Posson Dstrbutons.04 S00

12 Eamples of Accdental Rates Suppose we know λ, e.g. rate and therefore the mean λt Probablty of r λt λt e P r r! and the probablty of 0 λ λt e P0 0! λt e λt and probablty of fndng s P P0 λt Eample from PET : λ sngle rate R t resolvng tme τ So f wehave two rates, R R R τ of havng sgnals from both n the tme R acc fndng anevent nsome tme t t fndng eactly 0 n tme nterval of t and R then the probablty wndow s.04 S00

13 .04 S00 Gaussan Dstrbuton n n n n FWHM e P.35 ln π

14 Gaussan Dstrbuton.04 S00

15 Defnton of Full Wdth Half Mamum.04 S00

16 Confdence level vs. Standard Devaton.04 S00

17 .04 S00 Ch-square Dstrbuton ν ν ν ν ν χ ν freedom of degrees gamma functon Γ Γ u u n e u P u

18 What s meanng of c? Each term n the sum s just the devaton of from ts theoretcal mean dvded by the epected dsperson. Now, f the data ponts really are Gaussans, then the rato should be around, and the value of χ ν, the number of degrees of freedom. So, f our hypothess s reasonable, then we would see χ ν. If however a very dfferent value of χ s obtaned, then the orgnal hypothess may be questonable..04 S00

19 Ch-square and Goodness of Ft.04 S00

20 Sgnfcance Levels.04 S00

21 Eample of Weghted Data.e. weght by.04 S00

22 .04 S00 Propagaton of Errors y f f y y f f y f y f + +, cov,

23 Count Rate Eample Take seres of mnuterunsandget Supposewe just measured onefve mnuterun But count rate s just 06 ±.04 S00

24 Null Results P0 e λt P λ λ 0 < λ 0 λ 0 ln CL T 0 Te λt dλ e λ 0 T confdence level.04 S00

25 Lookng for Rare Processes, Eample Lookng for rare dsease Look at 0 5 people over 0 years..see no cases Assumng 90% CL λ ln. 0.3/ yr Per person, f we have 0 5 people, we get per person: S00

26 Rare Events, Eample We have 00 nukes n US and have not seen a major falure n 40 years λ ln0. 4 per reactor year BUT there may be correlatons.04 S00

27 Nose/Dose n Images n SNR n det n det n n det det SNR.04 S00

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