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

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Presentation on theme: "Ludlum Measurements, Inc. User Group Meeting June 22-23, 2009 San Antonio, TX."— Presentation transcript:

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 :

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333  10 5555 222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 or less?

21 Lambda = 3 DiscreteCumulative xp(x) ∑p(x)

22 Poisson Distribution, Lambda = 3

23 Poisson Distribution, Lambda = 1.25

24 Gaussian Distribution Function

25  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)”.

26

27 Excel Function F(2) = NORMDIST(2, 0, 1, TRUE) = 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… K B = NORMINV((1-P FA )^(1/N),0,1) False Alarm Probability MeanStdDev

33

34

35 Computing Alarm Setpoint

36 Simplify and Divide by Time

37 …almost!

38 Final Form:

39 Slight detour … 2-sided distribution

40

41

42

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

51 A Look at Q-PASS


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