Statistics

Decay Probability

• Radioactive decay is a statistical process.

– Assume N large for continuous function

• Express problem in terms of probabilities for a single event.

– Probability of decay p

– Probability of survival q

– Time dependent

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Combinatorics

• The probability that n specific occurrences happen is the product of the individual occurrences.

– Other events don’t matter.

– Separate probability for negative events

• Arbitrary 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.

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Bernoulli Process

• Treat events as a discrete trials.

– N separate trials

– Trials independent

– Binary outcome of trial

– Probability same for all trials.

• This defines a Bernoulli process.

Typical Problem• 10 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?

Answer• Probability of 1 decay,

• And 3 arbitrary atoms decay from the 10 and 7 do not:

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Binomial Distribution

• The general form of the Bernoulli process is the binomial distribution.

– Terms same as binomial expansion

• Probabilities are normalized.

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mathworld.wolfram.com

Mean and Standard Deviation

• The mean of the binomial distribution:

• Consider an arbitrary x, and differentiate, and set x = 1.

• The standard deviation of the binomial distribution:

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Disintegration Counts

• In counting experiments there is a factor for efficiency .

– Probability that a measurement is recorded

Typical Problem• A sample has 10 atoms of 42K

in an experiment with = 0.32. What is the expected count rate over 3 h?

Answer• Use the mean of the observable

count, convert to rate.

• 10(0.32)(0.154)/3 h = 0.163 h-1.

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More Counts

• Consider a source of 42K with an activity of 37 Bq, in a counter with = 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 is

• Decay constant is = ln2 / T = 0.056 h-1 = 1.55 x 10-5 s-1.

– The probability of decay is

• Number of atoms is N = rd / = 2.4 x 106.

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Poisson Distribution

• Many processes have a a large pool of possible events, but a rare occurrence for any individual event.

– Large N, small n, small p

• This is the Poisson distribution.

– Probability depends on only one parameter Np

– Normalized when summed from n =0 to .

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Poisson Properties

• The mean and standard deviation are simply related.

– Mean = Np, standard deviation 2 = ,

• Unlike the binomial distribution the Poisson function has values for n > N.

Poisson Away From Zero

• The Poisson distribution is based on the mean = Np.

– Assumed N >> 1, N >> n.

• Now assume that n >> 1, large and Pn >> 0 only over a narrow range.

• This generates a normal distribution.

• Let x = n – .

• Use Stirling’s formula

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Normal Distribution

• The full normal distribution separates mean and standard deviation parameters.

• Tables provide the integral of the distribution function.

• Useful benchmarks:

– P(|x - | < 1 = 0.683

– P(|x - | < 2 = 0.954

– P(|x - | < 3 = 0.997

Typical Problem• Repeated counts are made in 1-

min intervals with a long-lived source. The observed mean is 813 counts with = 28.5 counts. What is the probability of observing 800 or fewer counts?

Answer• This is about -0.45.

• Look up P((x-)/ < -0.45)

– P = 0.324

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Cumulative Probability

• Statistical processes can be described for large numbers.

– Can we model one event?

– No two events are equal

• Probability distributions typically reflect incidence in an infinitessimal region.

– Integrate over a range

• Consider an event with a 500 keV incident photon on soft tissue with attenuation = 0.091 cm-1.

• The probability of an interaction in 2 cm is

• P = 1 – 0.834 = 0.166

• How does one simulate this?

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Random Numbers

• To 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 starts

• Linear Congruential Generator

• Start with a seed value, X0.

– Select integers a, b.

• For a given Xn,

– Xn+1 = (aXn + b) mod m

• The maximum number of random values is m.

Random Distribution

• A selected random number is usually generated in a large range of integers.

– Uniform over the range

• Normalize values to select a narrow range.

– Usually from 0 to 1

• Convert 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

– (Box-Mueller algorithm)

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Monte Carlo Method

• The 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 .

– 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.

Interaction Simulation

Typical Problem• A 100 keV neutron beam is

incident on a mouse (3 cm thick). Calculate the energy deposited at different depths.

Data Table• H =0.777 cm-1

• O =0.100 cm-1

• C =0.0406 cm-1

• N =0.00555 cm-1

• Total =0.92315 cm-1

• Simulate one neutron.

• Find the distance of penetration by inverting the probability.

• Find the nucleus struck.

– Normalize the i to the total possible tot.

• Select the energy of recoil and angle.

• Repeat for new distance.

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Fitting Tests

• Collected data points will approximate the physical relationship with large statistics.

• Limited statistics require fits of the data to a functional form.

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.

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Polynomial Fit

• The procedure for a least squares fit applies to any polynomial.

– n+1 parameters ak

• Minimize error expression Q.

• Requires simultaneous solutions to a set of n+1 equations.

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Exponential Fit

• The 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 values

– Taylor’s series to linearize

– Find hk that minimize Q

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Chi Squared Test

• Fitting is based on a limited statistical sample.

• A chi-squared test measures the data deviation from the fit.

– Normally distributed

– Mean k for k degrees of freedom

• Divide the sample into n classes with probabilities pi and frequencies mi.

• The test is

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