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

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.

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:

Binomial Distribution The general form of the Bernoulli process is the binomial distribution. Terms same as binomial expansion Probabilities are normalized. mathworld.wolfram.com

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:

Disintegration Counts In counting experiments there is a factor for efficiency e. Probability that a measurement is recorded Typical Problem A sample has 10 atoms of 42K in an experiment with e = 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.

More Counts Consider 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 is Decay constant is l = ln2 / T = 0.056 h-1 = 1.55 x 10-5 s-1. The probability of decay is Number of atoms is N = rd /l = 2.4 x 106.

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 .

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.

Poisson Away From Zero The 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

Normal Distribution The 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) = 0.683 P(|x - m| < 2 s) = 0.954 P(|x - m| < 3 s) = 0.997 Typical Problem Repeated 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? Answer This is about -0.45s. Look up P((x-m)/s < -0.45) P = 0.324

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 m = 0.091 cm-1. The probability of an interaction in 2 cm is P = 1 – 0.834 = 0.166 How does one simulate this?

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 m, s (Box-Mueller algorithm)

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 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.

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 m=0.777 cm-1 O m=0.100 cm-1 C m=0.0406 cm-1 N m=0.00555 cm-1 Total m=0.92315 cm-1 Simulate 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.

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.

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.

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

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