Poisson Statistics. Objectives

Transcription

1 by Dr. James E. Parks Departmet of Physics ad Astroomy 401 ielse Physics Buildig The Uiversity of Teessee Koxville, Teessee Copyright March, 2001 by James Edgar Parks* *All rights are reserved. o part of this publicatio may be reproduced or trasmitted i ay form or by ay meas, electroic or mechaical, icludig photocopy, recordig, or ay iformatio storage or retrieval system, without permissio i writig from the author. Objectives The objectives of this experimet are: (1) to study the Poisso probability distributio as it is applied to coutig statistics, (2) to study the Gaussia probability distributio as a special case of the Poisso distributio applied to measuremets with a large mea value, ad (3) to lear some statistical aalysis techiques usig a spreadsheet to aalyze, graph, ad compare results. Theory Radioactive decay is a radom process i which the emissio of radiatio depeds o the umber of atoms that ca decay ad a probability fuctio that is characteristic their atural lifetimes. The detectio of particles is radom ad ay two measuremets of the particles over equal periods of time will most likely be differet. For large umbers, the differece betwee the measuremets will be a small percetage. The probability of detectig a specific umber of evets for a give measuremet is give by the stadard ormal or Gaussia distributio as was studied i the experimet o statistical aalysis. However, for the measuremet of a small umber of evets, the probability distributio for detectig a specific umber of evets is differet ad is give by a Poisso distributio. For example, i a small period of time, the umber of particles that are detected may oly be about 5 o the average. A chage of 1 or 2 detected evets is a sigificat percetage of the average umber. For rare evets, the average umber detected might also be much less tha oe, The Poisso distributio applies to these measuremets ad is useful for determiig the probability of detectig a sigle evet or more tha oe evet i the same period. The Poisso distributio is a special case of the biomial distributio, similar to the Gaussia distributio beig a special case.

2 The Poisso distributio is give by P ( ) =! e (1) where P() is the ormalized probability that i a give time iterval evets will be observed ad is the average umber of evets that are observed whe may samples are take. P() is said to be a ormalized distributio so that whe it is summed over all possible values of, it is equal to uity. P() is ormalized to oe sample ad is therefore give by P ( ) = 1 (2) = 0 If M samples are measured with each yieldig some umber m j, (j=1 to M), the average umber for a sample will be give by = M j= 1 M m j (3) If some fractio of the values of m j have the same value i ad there are possible values of i, (i=1 to ), the M f = m (4) i i j i= 1 j= 1 where f i is the frequecy that the same umber i occurs. By makig this substitutio ito Equatio (3), ad The term = i= 1 M f f i i (5) i = i (6) i= 1 M fi M is just the probability of measurig a value i, P( i ), so that By droppig the subscripts = P( ) (7) i= 1 i i 2

3 = P( ). (8) Whe the average umber of evets is a large umber, Poisso s equatio is very difficult to deal with because of the large values of! For large values of, the biomial distributio is better characterized with a Gaussia distributio. Recall from the previous experimet o statistical aalysis that the Gaussia distributio is give by P 1 x = exp - σ 2π ( ) ( x-x ) 2 2σ 2 (9) where x is the mea value about which the distributio is cetered, σ is the stadard deviatio from the mea, ad P(x) is the ormalized probability of measurig a value x. P(x i ) is a cotiuous fuctio ad must be multiplied by a icremet i x, Δ x, i order for it to represet a fiite umber. Therefore, the mea value is give by i ( i) Δ. (10) x= x P x x The value of σ is give by the square root of the variace, σ 2, i=1 ( x x) i i= σ =. (11) 1 Sice, the distributio is ormalized, P( xi ) Δ x = 1. (12) i=1 For itegers, the smallest icremet i x, Δ x =Δ, is 1, so whe the cotiuous fuctio P(x) is chaged to a iteger form P(), Equatio (9) becomes ad 1 P( ) Δ= exp - σ 2π 1 P ( ) = exp - σ 2π ( -) 2 2σ 2 ( -) 2 2σ 2 Δ. (13). (14) sice Δ = 1. The stadard deviatio ca be approximatedy by the square root of the mea, 3

4 ad Equatio (14) ca be approximated by σ =, (15) 1 P ( ) = exp - 2π ( -) 2 2. (16) The stadard deviatio ca be computed with Equatio (11). The stadard deviatio computatio is provided as a statistical fuctio i the Excel fuctioal operators ad the Gaussia distributio ca be computed easily ad more correctly as will be show. Apparatus The apparatus is show i Figure 1 ad cosists of: (1) a radioactive source, (2) a Geiger-Mueller tube with stad ad source holder, (3) a scaler to cout the radioactive decays, (4) a Pasco 750 Sciece Workshop data acquisitio iterface, (5) Pasco DataStudio software, ad (6) computer with Excel spreadheet program. Figure 1. Apparatus ad setup for performig the Poisso Statistics experimet. Procedure Geeral Cosideratios The procedure has two parts: (1) measuremets where the average umber of couts is low, approximately 2-7 couts per period, ad (2) measuremets where the average umber of couts is 100 to 200 couts per period. I the first case Poisso statistics apply, but i the secod case, a Gaussia distributio is applied. 4

5 The performace of the Geiger couter depeds o the high voltage supply ad that has bee pre-set for this experimet. ormally, a experimet would be ru to determie the operatig voltage, but that process has bee elimiated for this experimet to cocetrate o the study of the coutig statistics, the Poisso distributio, ad Gaussia distributio. Uless istructed otherwise, the high voltage for the couter should be set at 900 volts. The umber of radioactive evets that are detected by the Geiger couter depeds o the geometrical relatio that the source has to the couter. The closer the source is to the couter, the more evets will be detected. The source ca be located at various distaces from the etrace widow of the Geiger tube, ad this distace will eed to be adjusted to give the desired coutig rate for the study. Also, because of the locatio of the radioactive source material embedded i the plastic disc,the orietatio of the source will affect the cout rate.. There will be a differece i detected evets depedig o which side of the disc is closest to the Geiger widow. I Part I, the source will eed to be placed ear the bottom of the source holder ad will eed to be moved closer to the tube for Part II. The umber of couts that are detected i a 0.2 secod time iterval i Part I eeds to be approximately 2-7, while for Part II the umber eeds to be approximately Part I Poisso Distributio Data Acquisitio Setup 1. From the desktop ope the DataStudio program by clickig o the DataStudio ico. 2. Select Create a Experimet from the choices i the Welcome to DataStudio pop-up widow. 3. I the Experimet Setup widow scroll through the list of Sesors i the pull-dow meu to the Geiger Couter sesor ico,. Double click o the Geiger Couter sesor ico to add it to Digital Chael 1. Check ad make sure that the cable coected to the ucleus scaler is plugged ito Digital Chael 1 o you 750 iterface. 4. Double click o the Geiger Couter sesor ico coected to the 750 iterface i the Experimet Setup widow. This brigs up the Sesor Properties widow. Uder the geeral tab, click o the Fast [> 1 Hz] butto to select it (if it is t already selected), ad chage the sample rate to 5 Hz usig the +/- buttos,. This settig will set the data acquisitio system to cout sigal pulses durig 0.2 secod periods. Click o the Measuremet tab i this widow. Make sure that the Geiger Couuts, Ch1 (couts/sample check box is checked, ad the click OK. 5. Click o the Optios butto o the Setup Toolbar ad this will brig up the Samplig Optios widow. This widow ca be used to cotrol the startig ad stoppig of data collectio. I the Delay Start sectio, click the radio butto to select oe. 5

6 6. I the automatic stop sectio, select the radio butto Time ad set the time to secods. This will allow 2500 measuremets of the umber of couts that occur i 0.2 secod icremets. 7. Uder the Displays widow double click both o the graph ico ad the table ico. This will display the data collected from the Geiger couter i a table ad graph. 8. Arrage the graph ad table widows to best display your results. You ca hide the Displays ad Data widows by clickig o the small up ad dow arrow beside the Summary butto o the DataStudio toolbar. 9. Whe ready, click the Start butto to start collectig the data. The data will be recorded i the table ad graphed o the graph. If for some reaso data collectio does t stop, you ca stop data collectio by clickig o the Stop butto. 10. O each of the widow s Graph ad Table there is a drop dow meu beside the toolbar butto labeled Data. Select Clear Data from the drop dow meu. This will allow you to ru the experimet agai. Each data set is saved separately, so you ca ru as may trials as you wat to ru so that you ca the go back ad compare your results. Acquisitio of Data for Part I 1. Place the source i the source holder ear the bottom. Tur o the scaler with the Power rocker switch. Place the scaler i the stopped mode by depressig the Stop rocker switch ad the depressig the Reset rocker switch. 2. Click o the Start butto of DataStudio ad observe the umber of evets detected i each of the 0.2 secod time itervals. Move the source to a positio that yields a average umber equal to 2-7 couts. 3. Click o the Stop butto to stop the data acquisitio. Select Experimet from the DataStudio mai meu bar ad the Delete All Data Rus from the pull dow meu. Click o OK ad the experimet is set to take data. 4. Click o the Start butto ad collect data for the ext 500 secods. Observe the data as it is beig accumulated i the table. If the data collectio is t meetig the criteria that is desired, stop the experimet, clear out the results, readjust the parameters, ad begi agai. The accumulatio of data should stop automatically. Data Aalysis for Part I 1. Start the Excel spreadsheet program either through the start meu or with a shortcut o the desktop. 6

7 2. Retur to the DataStudio program, ad usig the left mouse butto, click o the area of Table 1 that has the title Geiger Couts, Ch1 ad Ru #x. This will highlight i yellow all the umbers i the table. (As a alterative, click o the uppermost of the cells i the left colum of time data ad while holdig the left mouse butto dow, drag the cursor to the lowermost cell cotaiig data i the right, couts, colum to select the etire data set of Geiger couts versus time. The select the Edit meu optio from the top meu bar of the DataStudio program ad the select Copy. 3. Retur to Sheet 1 of the Excel program ad locate your cursor i cell A1 ad click o this cell with the left butto of your mouse. Choose the Edit optio from the top meu bar ad the select Paste. These operatios should place your data from your ru ito the spreadsheet. Examie the copied data to make sure that this has bee doe. Your spreadsheet should look somethig like the oe i Figure 2. Figure 2. Example Excel spreadsheet for the recordig ad aalysis of Poisso statistics data. 4. Add the colum headigs show i the example for colums C, D, E, ad F. These should be labeled, Average,, Frequecy Distributio, ad Poisso Distributio. Format the colum widths by selectig colums A through F ad choosig Format from the Excel mai meu bar. Choose Colum ad Width... from the pull dow meus ad type 15 i the Colum Width s popup widow iput box. Select the block of cells A1:F2 ad choose Format from the Excel mai meu bar. Choose Cells... from the pull dow meu ad click o the Aligmet tab i the Format Cells popup widow. Click o the check box to choose the Wrap text optio. 5. I cell C3 fid the average cout for the 2500 measuremets. To do this, choose Isert from the mai meu bar ad the choose Fuctio. From the Paste Fuctio widow choose Statistics from the Fuctio category list ad AVERAGE from the Fuctio ame list. The click o OK. Iput B3:B2502 ito the umber 1 iput box ad click o OK. You ca check this result by summig all the umbers i the B 7

8 colum ad dividig by the umber of umbers. The average should be less tha 5 i order to provide a good test for the Poisso distributio. 6. I colum D, uder the label, isert umbers 0, 1, 2, 3,..., 20. You ca eter 0 i cell D3 ad isert a equatio i cell D4, ad the use the copy ad fill fuctios to eter the rest of the umbers. 7. I cells E3 to E23 fid the frequecy (the umber of times) each of the umbers appear i 2500 measuremets. I other words, how may of the measuremets resulted with a 0 cout, how may resulted i 1 cout, how may resulted i 2 couts, etc. To do this it is helpful to use a subtle feature of the Excel spreadsheet that uses arrays to eter ad extract data. Arrays must be etered i a differet maer tha other etries. To begi, click o cell E3 ad while holdig the left mouse butto dow, drag the cursor dow to cell E23 so that cells E3 to E23 are selected. Choose Isert from the mai meu bar ad the choose Fuctio. From the Paste Fuctio widow choose Statistics from the Fuctio category list ad FREQUECY from the Fuctio ame list. The click o OK. Iput B3:B2502 ito the Data array iput box ad D3:D23 i the Bis array iput box. While holdig the Ctrl ad Shift keys dow, click o OK. umbers should appear i all of cells D3 through D23. Check to see if these umbers seem reasoable. 8. Colum E should ow cotai the umber of times the measuremets produced the value of listed i colum D, i.e. colums D ad E should cotai the data for the frequecy distributio. A graph of the frequecy distributio should ow be made. 9. Select the data to be graphed i colums D ad E. Choose Isert ad the Chart. Choose XY (Scatter) as the Chart type ad the optio of Scatter. Compares pairs of values. for the Chart sub-type,. You ca click o ext ad supply the iformatio to format your graph i a meaigful way or just press Fiish ad go back later to format your graph. You should make sure the Series data assigs the values i the D colum as the values ad the values i the E colum as the Frequecy values. 10. Check your graph to make sure it is correct. It should look somethig like the oe show i Figure 3. The frequecy of umbers should be o the y-axis versus the umbers o the x-axis. Click o the graph ad hold the mouse butto dow to drag it to the desired locatio o your spreadsheet. 11. Add a title to the graph ad label the axes This ca be doe by clickig o Chart o the top meu bar ad the choosig the Chart Optios... selectio ad supplyig the iformatio. Save your Excel workbook frequetly to prevet loss of your work. 8

9 Measured Frequecy Distributio of Geiger Couts 600 Frequecy of per 2500 Samples Frequecy Distributio Figure 3. Example of graph of Poisso statistic data. 12. Test the data to check that it follows the Poisso distributio give by Equatio (1), P ( ) =! e. 13. I cell F3, eter the formula, =$C$3^D3/FACT(D3)*EXP(-$C$3)*2500. Recall that P() is a ormalized distributio ad that = 0 P ( ) = 1. Therefore each value of P() that is computed must be multiplied by 2500 to correspod to the frequecy distributio of 2500 measuremets. 14. Copy the formula i cell F3 ad copy it to the other cells, F4 through F Compare the values of the frequecy distributio i colum E with those of the Poisso distributio computed i colum F. The values should be about the same. 16. Add the data for the Poisso distributio to your graph. To do this click o your chart ear its edges to select it ad the select Chart from the mai meu bar. Choose Add Data... from the drop dow meu ad iput =Sheet1!F3:F23 ito the Rage iput box. Click o OK ad this should add the calculated values for the Poisso distributio to your graph. Your graph should look somethig like the oe i Figure Examie the Frequecy ad Poisso distributios ad determie how well they agree. 9

10 Frequecy Distributio ad Poisso Distributio of Geiger Couts 600 Frequecy of per 2500 Samples Frequecy Distributio Poisso Distributio Figure 4. Graph showig both the Frequecy distributio ad the Poisso distributio. Part II - Gaussia Distributio Acquisitio of Data for Part II 1. Move the source closer to the Geiger tube to icrease the cout rate. Click o the Start butto of DataStudio ad observe the umber of evets detected i each of the 0.2 secod time itervals. Adjust the source to a positio that yields a average umber equal to couts. 2. Click o the Stop butto to stop the data acquisitio. Select Experimet from the DataStudio mai meu bar ad the Delete All Data Rus from the pull dow meu. Click o OK ad the experimet is agai set to take data. 3. Click o the Start butto ad collect data for the ext 500 secods. Observe the data as it is beig accumulated i the table. If the data collectio is t meetig the criteria that is desired, stop the experimet, clear out the results, readjust the parameters, ad begi agai. The accumulatio of data should stop automatically. Data Aalysis 1. Ope the Excel workbook used i Part I ad go to a ew worksheet, Sheet 2. 10

11 2. Retur to the DataStudio program, copy the data i Table 1, ad paste it ito the Excel spreadsheet. Examie the copied data to make sure that this has bee doe. Your spreadsheet should look somethig like the oe i Figure 5. Figure 5. Example Excel spreadsheet for Gaussia distributio data. 3. Add the colum headigs show i the example for colums C, D, E, F, ad G. These should be labeled, Average,, Frequecy Distributio, Gaussia Distributio usig Approximatio, ad Gaussia Distributio usig Stadard Deviatio. Format the colum widths to a width of 15. Ceter the labels ad etries i the cells ad use the wrap text optio for the labels. 4. I cell C3 fid the average cout for the 2500 measuremets. The average should be i order to provide a good test for the Gaussia distributio. 5. Type the text Miimum ad Maximum i cells C5 ad C8 respectively. I cell C6 eter the fuctio for fidig the miimum cout for all 2500 measuremets, =MI(B3:B2502), ad i cell C9 eter the fuctio for fidig the maximum cout for all 2500 measuremets, =MAX(B3:B2502). This will give lower ad upper limits o the rage of values measured ad will elimiate graphig a large umber of zeros. 6. I cell D3, uder the headig, isert a iteger that is approximately 50 less tha a iteger close to the earest iteger value to the average cout for the 2500 measuremets. Eter the formula =D3+1 i cell D4 ad use the fill fuctio to copy this formula i cells D5 to D103. Check to see if the rage of values betwee the miimum ad maximum values for is covered by these umbers i colum D. If ot, add umbers to the begiig ad ed to cover the rage. 11

12 7. I cells E3 to E103 fid the frequecy (the umber of times) each of the umbers appear i 2500 samples. Remember that arrays must be etered i a differet maer tha other etries. Click o cell E3 ad select the cells dow to cell E103. Type =FREQUECY(B3:B2502,D3:D103) ad while holdig the Ctrl ad Shift keys dow, hit the Eter key. 8. Colum E should ow cotai the umber of times the measuremets produced the value of listed i colum D, i.e. colums D ad E should cotai the data for the frequecy distributio. A graph of the frequecy distributio should ow be made. 9. Select the data to be graphed i colums D ad E. Choose Isert ad the Chart. Choose XY (Scatter) as the Chart type ad the optio of Scatter. Compares pairs of values. for the Chart sub-type,. You ca click o ext ad supply the iformatio to format your graph i a meaigful way or just press Fiish ad go back later to format your graph. You should make sure the Series data assigs the values i the D colum as the values ad the values i the E colum as the Frequecy values. 10. Check your graph to make sure it is correct. It should look somethig like the plot of data poits show i Figure 6. The frequecy of umbers should be o the y-axis versus the umbers o the x-axis. Click o the graph ad hold the mouse butto dow to drag it to the desired locatio o your spreadsheet. 11. Add a title to the graph ad label the axes This ca be doe by clickig o Chart o the top meu bar ad the choosig the Chart Optios... selectio ad supplyig the iformatio. Save your Excel workbook frequetly to prevet loss of your work. 12. Check the data to see if it follows the Gaussia distributio give by Equatio (16), 1 P( ) = exp - 2π where σ has bee approximated by distributio ad that = 0 ( -) 2 2. Recall that P() is a ormalized P ( ) = 1. Therefore each value of P() that is computed must be multiplied by 2500 to correspod to the frequecy distributio of 2500 measuremets. I cell F3, eter the formula, =(1/(SQRT(2*PI()*$C$3)))*EXP(-((D3-$C$3)^2)/(2*$C$3))*

13 ad copy it to the other cells, F4 through F103. Frequecy Distributio ad Gaussia Distributio Frequecy of per 2500 Samples Measured Frequecy Gaussia Distributio w/ Approximatio Gaussia Distribuitio w/ Figure 6. Example graph showig the distributio of data poits ad the theoretical fits of the the Gaussia distributio to the data. 13. Compare the values of the frequecy distributio i colum E with those of the Gaussia distributio computed i colum F. The values should be about the same. 14. Add the data for the Gaussia distributio to your graph. To do this click o your chart ear its edges to select it ad the select Chart from the mai meu bar. Choose Add Data... from the drop dow meu ad iput =Sheet2!F3:F103 ito the Rage iput box. Click o OK ad this should add the calculated values for the Gaussia distributio to your graph. 15. Examie the measured frequecy distributio ad the calculated Gaussia distributios with the σ = approximatio ad determie how well they agree. 16. I cell C12 compute the stadard deviatio of the 2500 samples. To do this, isert the formula =STDEV(B3:B2502). STDEV is a statistical fuctio that ca be iserted from the Isert optio o the mai meu bar followed by choosig the Fuctio selectio. The stadard deviatio fuctio is available from the Fuctio ame list associated with the Statistical selectio uder the Fuctio category list. 17. Calculate the proper Gaussia distributio usig the equatio for the Gaussia distributio usig the stadard deviatio of the samples, 13

14 I cell G3, eter the formula, 1 P() = exp - σ 2π ( -) 2 2σ 2. (5) =(1/($C$12*SQRT(2*PI())))*EXP(-((D3-$C$3)^2)/(2*$C$12^2))*2500 ad copy it to the other cells, G4 through G103. Remember that P() is multiplied by 2500 sice P() is a ormalized distributio where P ( i ) = Add the data for this Gaussia distributio to your graph. To do this click o your chart ear its edges to select it ad the select Chart from the mai meu bar. Choose Add Data... from the drop dow meu ad iput =Sheet2!G3:G103 ito the Rage iput box. Click o OK ad this should add this Gaussia distributio to your graph. 19. Compare this distributio with the measured frequecy distributio ad the Gaussia distributio usig the σ = approximatio. Which oe is the best fit to the measuremets? CAUTIO: READ THE FOLLOWIG ISTRUCTIOS VERY CAREFULLY BEFORE ATTEMPTIG TO PRIT YOUR SPREADSHEETS AD GRAPHS. OLY A PORTIO OF THE SPREADSHEETS WILL EED TO BE PRITED. OTHERWISE, THE SPREADSHEETS REQUIRE MORE THA 50 PAGES TO BE PRITED. 20. Prit the 2 spreadsheets ad the 2 charts i your Excel workbook usig the followig istructios: For each spreadsheet that you eed to prit, you will oly eed to prit a portio of the spreadsheet. Pritig the etire spreadsheet will take a very log time ad will cosume more tha 50 sheets of paper. You should oly prit the first page. To prit oly the first page, select File from the mai meu bar, ad Prit... from the pull dow meu. I the Prit Rage sectio, select the Page(s) optio ad eter 1 i the From: iput box ad 1 i the To: iput box. Click o Preview to check to see if more tha oe page is set for pritig. If oly oe page shows up i the preview, the hit Prit ad the Close the preview. If more tha oe page shows up i the preview, the hit Close to close the preview to readjust the parameters. Charts may be prited by selectig the chart ad the choosig Prit... from the pull dow meu uder File from the mai meu. i= 0 14

15 Questios 1. The Stigy Salt Shaker Problem: Suppose after 1000 shakes of a stigy salt shaker, the average umber of grais of salt that comes out is 0.1 grai per shake. What is the probability that 2 grais of salt will come out for a give sigle shake? 2. O the average, there are about 2 major airlie crashes per year. What is the probability that i ay moth, 2 crashes will occur? 15