1. Basic Counting StatisticsBasic Counting StatisticsBasic Counting StatisticsBasic Counting Statistics

2. W. Udo Schröder, 2009 Statistics 2 Stochastic Nuclear Observables 2 sources of stochastic observables x in nuclear science: 1) Nuclear phenomena are governed by quantal wave functions and inherent statistics 2) Detection of processes occurs with imperfect efficiency (  < 1) and finite resolution distributing sharp events x 0 over a range in x. Stochastic observables x have a range of values with frequencies determined by probability distribution P(x). characterize by set of moments of P = ∫ x n P (x)dx; n 0, 1, 2,… Normalization = 1. First moment (expectation value) of P: E(x) = = ∫xP (x)dx second central moment = “variance” of P(x):  x 2 = >

3. Uncertainty and Statistics Nuclear systems: quantal wave functions  i (x,…;t) (x,…;t) = degrees of freedom of system, time. Probability density (e.g., for x, integrate over other d.o.f.) 1 2 Partial probability rates  for disappearance (decay of 1  2) can vary over many orders of magnitude  no certainty  statistics

4. W. Udo Schröder, 2009 Statistics 4 The Normal Distribution Continuous function or discrete distribution (over bins) Normalized probability 2x2x P(x) x x

5. W. Udo Schröder, 2009 Statistics 5 Experimental Mean Counts and Variance Measured by ensemble sampling expectation values + uncertainties Sample (Ensemble) = Population instant 236 U (0.25mg) source, count  particles emitted during N = 10 time intervals (samples @1 min).  =?? Slightly different from sample to sample

6. W. Udo Schröder, 2009 Statistics 6 Sample Statistics x P(x) Assume true population distribution for variable x with true (“population”) mean pop = 5.0, x =1.0 Mean arithmetic sample average = (5.11+4.96+4.96)/3 = 5.01 Variance of sample averages  s 2 =  2 = [(5.11-5.01 )2 +2(4.96-5.01) 2 ]/2 = 0.01 s  = 0.0075 pop  5.01 ± 0.05  x 2 = 0.0075/3 = 0.0025  x = 0.05  Result: pop  5.01 ± 0.05 -  =5.11  =1.11 =4.96  =0.94 +  =4.96  =1.23 3 independent sample measurements (equivalent statistics): x

7. W. Udo Schröder, 2009 Statistics 7 Example of Gaussian Population x m x m Sample size makes a difference (  weighted average) n = 10 n = 50 The larger the sample, the narrower the distribution of x values, the more it approaches the true Gaussian (normal) distribution.

8. W. Udo Schröder, 2009 Statistics 8 Central-Limit Theorem Increasing size n of samples: Distribution of sample means  Gaussian normal distrib. regardless of form of original (population) distribution. The means (averages) of different samples in the previous example cluster together closely.  general property:  The average of a distribution does not contain information on the shape of the distribution. The average of any truly random sample of a population is already somewhat close to the true population average. Many or large samples narrow the choices: smaller Gaussian width  Standard error of the mean decreases with incr. sample size

9. W. Udo Schröder, 2009 Statistics 9 Integer random variable m = number of events, out of N total, of a given type, e.g., decay of m (from a sample of N ) radioactive nuclei, or detection of m (out of N ) photons arriving at detector. p = probability for a (one) success (decay of one nucleus, detection of one photon) Choose an arbitrary sample of m trials out of N trials p m = probability for at least m successes (1-p) N-m = probability for N-m failures (survivals, escaping detection) Probability for exactly m successes out of a total of N trials How many ways can m events be chosen out of N ?  Binomial coefficient Total probability (success rate) for any sample of m events: Binomial Distribution

10. W. Udo Schröder, 2009 Statistics 10 Moments and Limits Distributions for N=30 and p=0.1  p=0.3 Poisson  Gaussian Probability for m “successes” out of N trials, individual probability p

11. W. Udo Schröder, 2009 Statistics 11 Poisson Probability Distribution Probability for observing m events when average is =  m=0,1,2,…  2 =   = = N·p and  2 =  is the mean, the average number of successes in N trials. Observe N counts (events)  uncertainty is  = √  Unlike the binomial distribution, the Poisson distribution does not depend explicitly on p or N ! For large N, p: Poisson  Gaussian (Normal Distribution) Results from binomial distribution in the limit of small p and large N (N·p > 0)

12. W. Udo Schröder, 2009 Statistics 12 Moments of Transition Probabilities Small probability for process, but many trials (n 0 = 6.38·10 17 )    0< n 0 · < ∞ Statistical process follows a Poisson distribution: n=“random” Different statistical distributions: Binomial, Poisson, Gaussian

13. W. Udo Schröder, 2009 Statistics 13 Radioactive Decay as Poisson Process Slow radioactive decay of large sample Sample size N » 1, decay probability p « 1, with 0 < N·p <  137 Cs unstable isotope  decay t 1/2 = 27 years  p = ln2/27 = 0.026/a = 8.2·10 -10 s -1  0 Sample of 1  g: N = 10 15 nuclei (=trials for decay) How many will decay(= activity  ) ?  = = N·p = 8.2·10 +5 s -1 Count rate estimate =d /dt = (8.2·10 +5 ± 905) s -1 estimated Probability for m actual decays P ( ,m) =

14. W. Udo Schröder, 2009 Statistics 14 Functions of Stochastic Variables Random independent variable sets {N 1 }, {N 2 },….,{N n } corresponding variances  1 2,  2 2,….,  n 2 Function f(N 1, N 2,….,N n ) defined for any tuple {N 1, N 2,….,N n } Expectation value (mean) Gauss’ law of error propagation: Further terms if N i not independent (  correlations) Otherwise, individual component variances (  f) 2 add.

15. W. Udo Schröder, 2009 Statistics 15 Example: Spectral Analysis Background B Peak Area A B1B1 B2B2 Analyze peak in range channels c 1 – c 2 : beginning of background left and right of peak n = c 1 – c 2 +1. Total area = N 12 =A+B N(c 1 )=B 1, N(c 2 )=B 2, Linear background = n(B 1 +B 2 )/2 Peak Area A: Adding or subtracting 2 Poisson distributed numbers N 1 and N 2 : Variances always add

16. W. Udo Schröder, 2009 Statistics 16 Confidence Level Assume normally distributed observable x: Sample distribution with data set  observed average and std. error  approximate population. Confidence level CL (Central Confidence Interval): With confidence level CL (probability in %), the true value differs by less than  = n  from measured average. Trustworthy exptl. results quote ±3  error bars! Measured Probability

17. W. Udo Schröder, 2009 Statistics 17 Setting Confidence Limits Example: Search for rare decay with decay rate  observe no counts within time  t. Decay probability law dP/dt=-dN/Ndt= exp {- ·t}. P( t ) = symmetric in  and t normalized P Upper limit Higher confidence levels CL (0  CL  1)  larger upper limits for a given time  t inspected. Reduce limit by measuring for longer period. no counts in  t

18. W. Udo Schröder, 2009 Statistics 18 Maximum Likelihood Measurement of correlations between observables y and x: {x i, y i | i=1-N} Hypothesis: y(x) =f(c 1,…,c m ; x). Parameters defining f: {c 1,…,c m } n dof =N-m degrees of freedom for a “fit” of the data with f. for every data point Maximize simultaneous probability Minimize chi-squared by varying {c 1,…,c m }: ∂  2 /∂c i = 0 When is   as good as can be?

19. W. Udo Schröder, 2009 Statistics 19 Minimizing  2 Example: linear fit f(a,b;x) = a + b·x to data set {x i, y i,  i } Minimize: Equivalent to solving system of linear equations

20. W. Udo Schröder, 2009 Statistics 20 Distribution of Chi-Squareds Distribution of possible  2 for data sets distributed normally about a theoretical expectation (function) with n dof degrees of freedom: (Stirling’s formula) Reduced  2 : ndof =5 u:=  2 Should be P  0.5 for a reasonable fit

21. W. Udo Schröder, 2009 Statistics 21 CL for  2 -Distributions 1-CL

22. W. Udo Schröder, 2009 Statistics 22 Correlations in Data Sets Correlations within data set. Example: y i small whenever x i small x y uncorrel. P(x,y) x y correlated P(x,y) 

23. W. Udo Schröder, 2009 Statistics 23 Correlations in Data Sets uncertainties of deduced most likely parameters c i (e.g., a, b for linear fit) depend on depth/shallowness and shape of the  2 surface cici cjcj uncorrelated  2 surface cici cjcj correlated  2 surface

24. W. Udo Schröder, 2009 Statistics 24 Multivariate Correlations cici cjcj initial guess search path Smoothed  2 surface Different search strategies: Steepest gradient, Newton method w or w/o damping of oscillations, Biased MC:Metropolis MC algorithms, Simulated annealing (MC derived from metallurgy),  Various software packages LINFIT, MINUIT,….