Ludlum Measurements, Inc.

1. Ludlum Measurements, Inc. User Group Meeting June 22-23, 2009 San Antonio, TX

2. Counting Statistics James K. Hesch Santa Fe, NM

3. Binary Processes • Success vs. Failure • Go or No Go • Hot or Not • Yes or No • Win vs. Lose • 1 or 0 • Disintegrate or not • Count a nuclear event or not

4. Uncertainty • Shades of gray – neither black nor white • How gray is gray? • More black than white, or more white than black?

5. Some Familiar Real World Applications

6. What is the probability of drawing a Royal Flush in five cards drawn randomly from a deck of 52 cards?

7. The first card must be a member of the set [10, J, Q, K, A] in any of the four suites. Thus it can be any one of 20 cards.

8. The set of valid cards diminishes to four for the second card out of the remaining 51 cards, etc.

9. Probability 1 : 649740

10. Plato’s Real vs. Ideal Worlds • Observed vs. Expected • Predicting with uncertainty • Science is inexact • Stating the precision • “+/- 2% at the 95% confidence level”

11. Toss of One Die

12. Toss of Two Dice

13. Four Tosses of a Pair of Dice • 3 • 10 • 5 • 2 • Total = 20 • Average (Mean) = 20/4 = 5 • Compute the average value by which each toss in this sample VARIES from the mean.

14. Variance = σ²

15. Toss of Three Dice

16. Toss of Four Dice

17. Probability Distribution Functions • Binomial • Poisson • Gaussian or Normal (the famous bell curve)

18. Binomial Distribution Function

19. Poisson Distribution Function

20. Sample Exercise In a counting exercise where the average number of counts expected from background is 3, what should the minimum alarm set point be to produce a false alarm probability of 0.001 or less?

21. Lambda = 3

22. Poisson Distribution, Lambda = 3

23. Poisson Distribution, Lambda = 1.25

24. Gaussian Distribution Function

25. Gaussian Distribution Function • Is a Density Function, or cumulative probability (as opposed to discreet). • Can use look-up table or Excel functions to apply • Scale to data by use of Mean and Standard Deviation • Single-sided confidence – but can be used to determine two-sided confidence function “Erf(x)”.

27. Excel Function F(2) = NORMDIST(2, 0, 1, TRUE) = 0.97725 2 StdDev Mean = 0 StdDev of Data = 1 Cumulative = True

28. If NORMDIST() set to FALSE…

29. Controlling False Alarm Probability • Determine expected number of background counts that would occur in a single count cycle. • Determine the StdDev of that value • Set the alarm setpoint a sufficient number of Standard Deviations above average background counts for an acceptable false alarm probability.

30. False Alarm Probability

31. How Many Sigmas?

32. In Excel… KB = NORMINV((1-PFA)^(1/N),0,1) False Alarm Probability Mean StdDev

35. Computing Alarm Setpoint

36. Simplify and Divide by Time

37. …almost!

38. Final Form:

39. Slight detour … 2-sided distribution

43. In Excel… • Two sided distribution… • …=2*(NORMDIST(x, 0, 1, TRUE) – 0.5)

44. Getting Back to Alarm Setpoint…

45. MDA-Driven Alarm Setpoint

46. “Minimum” Count Time • Solve for T using the simplified equation below, and round up to a full no. of seconds: • Compute a new value for MDA (see next slide) using the resulting “T” as • As needed, iteratively, add 1 second to the T and recompute MDA until the result is < the desired MDA

47. Computing MDA • Start with MDA=1 for the right side of the following equation, and compute a new value for MDA • Substitute the new value on the right hand side and repeat. • Continue with the substitution/computation until the value for MDA is sufficiently close to the previous value.

48. Activity Other than MDA

49. Approximation of Nuisance Alarms

50. With Extended Count Time