1. Statistics

2. 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

3. 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.

4. 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:

5. Binomial Distribution
The general form of the Bernoulli process is the binomial distribution.
Terms same as binomial expansion
Probabilities 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 recorded
Typical Problem
A sample has 10 atoms of 42K in an experiment with e = 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 = h-1.

8. 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 = 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.

9. 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 .

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 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

12. 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

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

14. 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.

15. 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)

16. 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.

17. 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= cm-1
N m= cm-1
Total m= 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.

18. Fitting Tests
Collected 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 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.

21. 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

22. 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