Ludlum Measurements, Inc.Ludlum Measurements, Inc.

User Group MeetingUser Group Meeting

June 22-23, 2009June 22-23, 2009

San Antonio, TXSan Antonio, TX

Counting StatisticsCounting Statistics

James K. HeschJames K. Hesch

Santa Fe, NMSanta Fe, NM

Binary ProcessesBinary Processes

Success vs. FailureSuccess vs. Failure Go or No GoGo or No Go Hot or NotHot or Not Yes or NoYes or No Win vs. LoseWin vs. Lose 1 or 01 or 0 Disintegrate or notDisintegrate or not Count a nuclear event or notCount a nuclear event or not

UncertaintyUncertainty

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

black?black?

Some Familiar Real World ApplicationsSome Familiar Real World Applications

What is the probability of drawing a What is the probability of drawing a Royal Flush in five cards drawn Royal Flush in five cards drawn

randomly from a deck of 52 cards?randomly from a deck of 52 cards?

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

of 20 cards.of 20 cards.

3846.052

20p

The set of valid cards diminishes to The set of valid cards diminishes to four for the second card out of the four for the second card out of the

remaining 51 cards, etc.remaining 51 cards, etc.

48

1

49

2

50

3

51

4

52

20p

Probability 1 : 649740Probability 1 : 649740

000001359.0!52

)!47)(!4(20p

Plato’s Real vs. Ideal WorldsPlato’s Real vs. Ideal Worlds

Observed vs. ExpectedObserved vs. Expected Predicting with uncertaintyPredicting with uncertainty Science is inexactScience is inexact Stating the precisionStating the precision “ “+/- 2% at the 95% confidence level”+/- 2% at the 95% confidence level”

Toss of One DieToss of One Die

Single Die Results Distribution

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1 2 3 4 5 6

Value

Fre

qu

ency

Toss of Two DiceToss of Two Dice

Two Dice Results Distribution

0.00%

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4.00%

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18.00%

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Value (Sum)

Fre

qu

ency

Four Tosses of a Pair of DiceFour Tosses of a Pair of Dice

33 1010 55 22 Total = 20Total = 20 Average (Mean) = 20/4 = 5Average (Mean) = 20/4 = 5 Compute the average value by which each Compute the average value by which each

toss in this sample VARIES from the mean.toss in this sample VARIES from the mean.

Variance = Variance = σσ²²

1

)(2

n

Xx

1

)(1

2

2

n

Xxn

ii

Toss of Three DiceToss of Three Dice

Three Dice Results Distribution

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5

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30

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Value (Sum)

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qu

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Toss of Four DiceToss of Four Dice

Four Dice Results Distribution

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Probability Distribution FunctionsProbability Distribution Functions

BinomialBinomial PoissonPoisson Gaussian or Normal (the famous bell curve)Gaussian or Normal (the famous bell curve)

Binomial Distribution FunctionBinomial Distribution Function

kNkp pp

kNk

N

k

NP

)1(

)!(!

!

Poisson Distribution FunctionPoisson Distribution Function

!)(

x

exp

x

Sample ExerciseSample Exercise

In a counting exercise where the average In a counting exercise where the average number of counts expected from background number of counts expected from background is 3, what should the minimum alarm set point is 3, what should the minimum alarm set point

be to produce a false alarm probability of be to produce a false alarm probability of 0.001 or less?0.001 or less?

Lambda = 3Lambda = 3

  Discrete Cumulative

x p(x)  ∑p(x)

0 0.04979 0.04979

1 0.14936 0.19915

2 0.22404 0.42319

3 0.22404 0.64723

4 0.16803 0.81526

5 0.10082 0.91608

6 0.05041 0.96649

7 0.02160 0.98810

8 0.00810 0.99620

9 0.00270 0.99890

10 0.00081 0.99971

11 0.00022 0.99993

12 0.00006 0.99998

Poisson Distribution, Lambda = 3Poisson Distribution, Lambda = 3

Poisson Distribution, Lambda = 3

0%

5%

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25%

0 1 2 3 4 5 6 7 8 9 10 11 12

Discrete Value, (x)

Pro

bab

ilit

y

Poisson Distribution, Lambda = 1.25Poisson Distribution, Lambda = 1.25

Poisson Distribution, Lambda = 1.25

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0 1 2 3 4 5 6 7 8 9 10 11 12

Discrete values of x

Gaussian Distribution FunctionGaussian Distribution Function

dxexPx

2

2

2

1)(

Gaussian Distribution FunctionGaussian Distribution Function

Is a Density Function, or cumulative Is a Density Function, or cumulative probability (as opposed to discreet).probability (as opposed to discreet).

Can use look-up table or Excel functions to Can use look-up table or Excel functions to applyapply

Scale to data by use of Mean and Standard Scale to data by use of Mean and Standard DeviationDeviation

Single-sided confidence – but can be used Single-sided confidence – but can be used to determine two-sided confidence function to determine two-sided confidence function “Erf(x)”.“Erf(x)”.

Gaussian Distribution Function

Excel FunctionExcel Function

F(2) = NORMDIST(2, 0, 1, TRUE) = 0.97725F(2) = NORMDIST(2, 0, 1, TRUE) = 0.97725

2 StdDev2 StdDevMean = 0Mean = 0

StdDev of Data = 1StdDev of Data = 1Cumulative = TrueCumulative = True

If NORMDIST() set to FALSE…If NORMDIST() set to FALSE…

Controlling False Alarm ProbabilityControlling False Alarm Probability

Determine expected number of background Determine expected number of background counts that would occur in a single count counts that would occur in a single count cycle.cycle.

Determine the StdDev of that valueDetermine the StdDev of that value Set the alarm setpoint a sufficient number of Set the alarm setpoint a sufficient number of

Standard Deviations above average Standard Deviations above average background counts for an acceptable false background counts for an acceptable false alarm probability.alarm probability.

False Alarm ProbabilityFalse Alarm Probability

NBFA KFP )(1

How Many Sigmas?How Many Sigmas?

)1(1 NFAB PFK

In Excel…In Excel…

KKBB = NORMINV((1-P = NORMINV((1-PFAFA)^(1/N),0,1))^(1/N),0,1)

False Alarm ProbabilityFalse Alarm ProbabilityMeanMean

StdDevStdDev

Computing Alarm SetpointComputing Alarm Setpoint

BBBBA NNKNN

T

NK

T

N BB

A

Simplify and Divide by TimeSimplify and Divide by Time

T

RKR BBMINA )(

……almost!almost!

Final Form:Final Form:

B

BBBA T

R

T

RKR (min)

Slight detour … 2-sided distributionSlight detour … 2-sided distribution

±σ

±1 StdDev = 68%

±2 StdDev = 95%

±3 StdDev = 99.7%

In Excel…In Excel…

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

Getting Back to Alarm Setpoint…Getting Back to Alarm Setpoint…

B

BBBSMAXA T

R

T

REffMDAKEffMDAR

)(

MDA-Driven Alarm SetpointMDA-Driven Alarm Setpoint

Maximum Alarm Set Point

““Minimum” Count TimeMinimum” Count Time

Solve for T using the simplified equation below, and round Solve for T using the simplified equation below, and round up to a full no. of seconds:up to a full no. of seconds:

Compute a new value for MDA (see next slide) using the Compute a new value for MDA (see next slide) using the resulting “T” as resulting “T” as

As needed, iteratively, add 1 second to the T and As needed, iteratively, add 1 second to the T and recompute MDA until the result is recompute MDA until the result is << the desired MDA the desired MDA

²

EffMDA

REffMDAKRKT BBSBB

Computing MDAComputing MDA

Start with MDA=1 for the right side of the following Start with MDA=1 for the right side of the following equation, and compute a new value for MDAequation, and compute a new value for MDA

Substitute the new value on the right hand side and repeat.Substitute the new value on the right hand side and repeat. Continue with the substitution/computation until the value Continue with the substitution/computation until the value

for MDA is sufficiently close to the previous value.for MDA is sufficiently close to the previous value.

Eff

T

R

T

REffMDAK

T

R

T

RK

MDA B

BBBS

B

BBB

Eff

T

R

T

REffMDAK

T

R

T

RK

MDA B

BBBS

B

BBB

Activity Other than MDAActivity Other than MDAAlarm Probability vs. Activity Level

0

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1

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Percent MDA

Ala

rm P

rob

abil

ity

Approximation of Nuisance AlarmsApproximation of Nuisance Alarms

Nuisance Alarms

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abil

ity

With Extended Count TimeWith Extended Count Time

Nuisance Alarms

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Ala

rm P

rob

abil

ity

A Look at Q-PASSA Look at Q-PASS1000 cps Background

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Elapsed Time

Clean High Alarm