PHYSICS 2150 LABORATORY

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1 PHYSICS 2150 LABORATORY Professor John Cumalat TAs: Adam Green John Houlton Lab Coordinator: Scott Pinegar Lecture 6 Feb. 17, 2015

2 ANNOUNCEMENT The problem set will be posted on the course website or you can pick one up. They are due Tuesday, Feb. 24 at 5pm. Place them in the black box in lab. Your second lab report is due on Friday, Feb.20. This is the last lecture in Physics 2150.

3 THIS LECTURE Introduction to the Poisson Distribution More comments on fitting data

4 COUNTING EXPERIMENTS Should there be a consistent trend? NO! The long half-life assures that over the time of the experiment, the decay rate isn t changing significantly. Should the five intervals all have the same number of decays? NO! The decay is a random process.

5 THE POISSON DISTRIBUTION This is obviously asymmetric, doesn t allow negative events, and is obviously discrete (factorial is only defined for nonnegative integers). Note that the mean µ doesn t have to be an integer. The sum rule works too: try it out at home. Note that µ n factor keeps probability down for too few events; n! factor keeps it down for too many events.

6 The Poisson Distribution, cont. What s the average? P µ ( n) = e µ µ n n! Where n! = n υ = µ n P µ (n) = n n e µ n! n =0 n =0 n n! = 1 ; n = 0 term is zero (n 1)! µ n 1 υ = µe µ = µe µ (1+ µ + µ2 (n 1)! 2! + µ3 3!...) n =1 υ = µ e µ

7 :Example: Cosmic Ray experiment in which a detector counts an average of 20 cosmic rays per day. How many times during a one year period would you expect the detector to count just 10 cosmic rays over a day? µ = 20 ; P µ (n) = e µ µ n n! (20) 10 (10) 10 (2) 10 P 20 (10) = e 20 = e 20 10! 10! = /day = e (# of expected 10 count days) = 365 ( ) = 2.12 expect to observe 10 events twice in one year!

8 The Poisson Distribution, cont. What s the standard deviation? σ 2 υ = ( υ υ ) 2 = υ 2 2υυ + ( υ ) 2 = υ 2 ( υ ) 2 ; but already shown ( υ ) = µ υ 2 = µ n 2 P µ (n) = n n 2 e µ n! n =0 n =0 nµ n 1 (m +1)µ m υ 2 = µe µ = µe µ = µe µ {e µ µ + e µ } (n 1)! (m)! n =0 n =0 υ 2 = µ 2 + µ σ υ 2 = µ ; σ µ = µ

9 The Poisson Distribution, possesses Reproductive Property If X and Y are independent, Poisson distributed random variables P X (x) = (X )x e X x! ; P Y (y) = (Y )y e Y then Z= X + Y is Poisson distributed random variable, with P Z (z) = (Z )z e Z z! y! ; where Z = X + Y Means can accumulate for an extended time period OR divide the time period into a number of smaller periods and get same result.

10 :Example: Suppose we make ten 1 minute counts x = x i N = = 20.1 cnts/min σ = x = 20.1 = 4.5 cnts/min σ x = σ N = x =1.4 cnts/min N x = (20.1±1.4) cnts/min 201

11 :Example, cont: If we had chosen to regard this as one 10 minute count x T = 201 cnts σ T = 201 =14.18 cnts and on a per minute basis x = (20.1 ± 1.4) cnts/min same as before 201

12 GAUSS VS. POISSON GAUSSIAN POISSON

13 POISSON DISTRIBUTION

14 POISSON DISTRIBUTION

15 SMALL-MEAN BEHAVIOR OF POISSON DISTRIBUTION

16 LARGE-MEAN POISSON BEHAVIOR: GAUSSIAN LIMIT

17 HOW TO USE POISSON DISTRIBUTIONS IN COUNTING Uncertainty: σ= µ (for large µ, can be interpreted similarly to gaussian σ) STATISTICAL uncertainty on the number of counts is thus the square root of the number of counts. Background processes may be contributing to your rate: µ=µsignal+µbkg. Often, can measure µbkg by turning off the signal, so when you then measure µtotal you can subtract the background to measure your signal rate. How do you handle the error in this situation?

18 Counting in Presence of Background Let s say we have a total count and a background count and we want the net count say radioactivity from mortar in fireplace. Let N=Net Count; T= total Count; B = background count N = T B ; σ N = σ T 2 +σ B 2 σ N = T + B N If T=200 and B= 150; N = 50 +/- (350).5 =50+/- 19

19 An Example Claim to have 160 accidents in a 10 year period at the intersection of Broadway and Tablemesa Drive. The City of Boulder decides to install a turn-only lane. Over the next two years they record 26 accidents at the intersection. The city makes the claim that the accident rate has gone down. Can they justify their claim statistically? (16 +/- 1.26) per year - (13.0+/- 2.55) per year = (3 +/- 2.8 ) accidents per year Yes it is significant!

20 A SCIENCE EXAMPLE: FITTING RADIOISOTOPE LIFETIME Measure count rate (background) with no radioactive material present Introduce radioactive material Count the number of decays in a 1-second period; remeasure every 30 seconds Subtract the background rate Fit the rate vs. time to an exponential lifetime

21 MEASURING BACKGROUND Background rate should be constant, so each trial is a remeasurement of the same thing Get the background rate by taking mean: 402/10=40.2 (note that this is exactly equivalent to simply measuring for 10 seconds and dividing by 10 to find the rate). Uncertainty in the rate: total counts is 402± 402 So in 10 seconds, mean = 402±20 Divide by 10 to get rate in 1 sec: 40.2±2.0 Background rate is 40.2±2.0 counts/second

22 SIGNAL DATA Are all the trials measuring the same rate? What are the uncertainties in the number of counts? What is the measured signal rate? Subtract background (40.2 cts/sec) Statistical error remains the square root of the total number of counts Also there is a systematic error (not shown) due to uncertainty in the background

23 FITTING FOR THE MEAN LIFE Plot the rate vs. seconds To fit to a line, take the log: N(t) = N0 exp( t/ ) ln(n) = ln(n0) t/ Linear fit: y= ln(n); x=t. What s the uncertainty on y if uncertainty on N is σn?

24 HOW GOOD IS THE FIT? Often want to know how good a fit is. Minimizing χ 2 told you what the best fit to your function was The value of that minimum χ 2 can tell you how well data actually fit your function.

25 THE CHI-SQUARED TEST in high-dof limit

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27 LEAST-SQUARES AND CHI- SQUARED If you do a straight-line fit without using errors (unweighted leastsquares fit) Fit returns error on y (assuming they are all the same) Min χ 2 /dof is 1 by definition Can t use χ 2 /dof to determine fit quality If you use externally known uncertainties to do a weighted line fit (See Taylor problems 8.9, 8.19) Fit returns function parameters (slope,offset) and χ 2 You can use the table to determine fit compatibility with data as a confidence probability

28 LOOK AT THE DECAY DATA

29 Final Comment Don t hesitate to ask for assistance on the homework. I want all of you to learn the material! Good Luck!