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

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Counting Statistics James K. Hesch Santa Fe, NM

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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

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

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Some Familiar Real World Applications

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

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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.

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

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Probability 1 : 649740

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

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Toss of One Die

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Toss of Two Dice

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Four Tosses of a Pair of Dice 3333 10 5555 2222 Total = 20 Average (Mean) = 20/4 = 5 Compute the average value by which each toss in this sample VARIES from the mean.

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Variance = σ²

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Toss of Three Dice

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Toss of Four Dice

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

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Binomial Distribution Function

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Poisson Distribution Function

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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?

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Lambda = 3 DiscreteCumulative xp(x) ∑p(x) 00.04979 10.149360.19915 20.224040.42319 30.224040.64723 40.168030.81526 50.100820.91608 60.050410.96649 70.021600.98810 80.008100.99620 90.002700.99890 100.000810.99971 110.000220.99993 120.000060.99998

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Poisson Distribution, Lambda = 3

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Poisson Distribution, Lambda = 1.25

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Gaussian Distribution Function

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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)”.

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

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If NORMDIST() set to FALSE…

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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.

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False Alarm Probability

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How Many Sigmas?

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In Excel… K B = NORMINV((1-P FA )^(1/N),0,1) False Alarm Probability MeanStdDev

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Computing Alarm Setpoint

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Simplify and Divide by Time

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…almost!

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Final Form:

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Slight detour … 2-sided distribution

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

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Getting Back to Alarm Setpoint…

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MDA-Driven Alarm Setpoint

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“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

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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.

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Activity Other than MDA

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Approximation of Nuisance Alarms

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With Extended Count Time

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A Look at Q-PASS

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