2 Decay Probability Radioactive decay is a statistical process. Assume N large for continuous functionExpress problem in terms of probabilities for a single event.Probability of decay pProbability of survival qTime dependent
3 CombinatoricsThe probability that n specific occurrences happen is the product of the individual occurrences.Other events don’t matter.Separate probability for negative eventsArbitrary choice of events require permutations.Exactly n specific events happen at p:No events happen except the specific events:Select n arbitrary events from a pool of N identical types.
4 Bernoulli Process Treat events as a discrete trials. N separate trials Trials independentBinary outcome of trialProbability same for all trials.This defines a Bernoulli process.Typical Problem10 atoms of 42K with a half-life of 12.4 h is observed for 3 h. What is the probability that exactly 3 atoms decay?AnswerProbability of 1 decay,And 3 arbitrary atoms decay from the 10 and 7 do not:
5 Binomial Distribution The general form of the Bernoulli process is the binomial distribution.Terms same as binomial expansionProbabilities are normalized.mathworld.wolfram.com
6 Mean and Standard Deviation The mean m of the binomial distribution:Consider an arbitrary x, and differentiate, and set x = 1.The standard deviation s of the binomial distribution:
7 Disintegration Counts In counting experiments there is a factor for efficiency e.Probability that a measurement is recordedTypical ProblemA sample has 10 atoms of 42K in an experiment with e = What is the expected count rate over 3 h?AnswerUse the mean of the observable count, convert to rate.10(0.32)(0.154)/3 h = h-1.
8 More CountsConsider a source of 42K with an activity of 37 Bq, in a counter with e = 0.32 measured in 1 s intervals.What is the mean count rate?What is the standard deviation of the count rate?The mean disintegration rate is just the activity, rd = 37 Bq.The count rate isDecay constant is l = ln2 / T = h-1 = 1.55 x 10-5 s-1.The probability of decay isNumber of atoms is N = rd /l = 2.4 x 106.
9 Poisson DistributionMany processes have a a large pool of possible events, but a rare occurrence for any individual event.Large N, small n, small pThis is the Poisson distribution.Probability depends on only one parameter NpNormalized when summed from n =0 to .
10 Poisson Properties The mean and standard deviation are simply related. Mean m = Np, standard deviation s2 = m,Unlike the binomial distribution the Poisson function has values for n > N.
11 Poisson Away From ZeroThe Poisson distribution is based on the mean m = Np.Assumed N >> 1, N >> n.Now assume that n >> 1, m large and Pn >> 0 only over a narrow range.This generates a normal distribution.Let x = n – m.Use Stirling’s formula
12 Normal DistributionThe full normal distribution separates mean m and standard deviation s parameters.Tables provide the integral of the distribution function.Useful benchmarks:P(|x - m| < 1 s) = 0.683P(|x - m| < 2 s) = 0.954P(|x - m| < 3 s) = 0.997Typical ProblemRepeated counts are made in 1-min intervals with a long-lived source. The observed mean is 813 counts with s = 28.5 counts. What is the probability of observing 800 or fewer counts?AnswerThis is about -0.45s.Look up P((x-m)/s < -0.45)P = 0.324
13 Cumulative Probability Statistical processes can be described for large numbers.Can we model one event?No two events are equalProbability distributions typically reflect incidence in an infinitessimal region.Integrate over a rangeConsider an event with a 500 keV incident photon on soft tissue with attenuation m = cm-1.The probability of an interaction in 2 cm isP = 1 – = 0.166How does one simulate this?
14 Random NumbersTo simulate a statistical process one needs a random selection from the possible choices.Algorithms can generate pseudo-random numbers.Uncorrelated over a large range of trials.Randomness limited for large sets or fixed startsLinear Congruential GeneratorStart with a seed value, X0.Select integers a, b.For a given Xn,Xn+1 = (aXn + b) mod mThe maximum number of random values is m.
15 Random DistributionA selected random number is usually generated in a large range of integers.Uniform over the rangeNormalize values to select a narrow range.Usually from 0 to 1Convert range to match a distribution.To select a number with a normal distribution:Take two random numbers R1, R2 from 1 to N.Apply algorithm with m, s(Box-Mueller algorithm)
16 Monte Carlo MethodThe Monte Carlo method simulates complicated systems.Use random numbers with distribution functions to select a value.Test that value to see if it meets certain conditions.Simple Monte Carlo for p.Select a pair of random numbers from 0 to 1.Sum the squares and count if it’s less than 1.Multiply the fraction that succeed by 4.
17 Interaction Simulation Typical ProblemA 100 keV neutron beam is incident on a mouse (3 cm thick). Calculate the energy deposited at different depths.Data TableH m=0.777 cm-1O m=0.100 cm-1C m= cm-1N m= cm-1Total m= cm-1Simulate one neutron.Find the distance of penetration by inverting the probability.Find the nucleus struck.Normalize the mi to the total possible mtot.Select the energy of recoil and angle.Repeat for new distance.
18 Fitting TestsCollected data points will approximate the physical relationship with large statistics.Limited statistics require fits of the data to a functional form.
19 Least Squares Assume that the data fits to a straight line. Use a mean square error to determine closeness of fit.Minimize the mean square error.
20 Polynomial FitThe procedure for a least squares fit applies to any polynomial.n+1 parameters akMinimize error expression Q.Requires simultaneous solutions to a set of n+1 equations.
21 Exponential FitThe least squares fit can be applied to other functions.For a single exponential a fit can be made on the log.For the sum of exponentials consider constants a, b.Select initial valuesTaylor’s series to linearizeFind hk that minimize Q
22 Chi Squared Test Fitting is based on a limited statistical sample. A chi-squared test measures the data deviation from the fit.Normally distributedMean k for k degrees of freedomDivide the sample into n classes with probabilities pi and frequencies mi.The test is