Presentation on theme: "Ludlum Measurements, Inc. User Group Meeting June 22-23, 2009 San Antonio, TX."— Presentation transcript:
Ludlum Measurements, Inc. User Group Meeting June 22-23, 2009 San Antonio, TX
Counting Statistics James K. Hesch Santa Fe, NM
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
Uncertainty Shades of gray – neither black nor white How gray is gray? More black than white, or more white than black?
Some Familiar Real World Applications
What is the probability of drawing a Royal Flush in five cards drawn randomly from a deck of 52 cards?
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.
The set of valid cards diminishes to four for the second card out of the remaining 51 cards, etc.
Probability 1 :
Plato’s Real vs. Ideal Worlds Observed vs. Expected Predicting with uncertainty Science is inexact Stating the precision “+/- 2% at the 95% confidence level”
Toss of One Die
Toss of Two Dice
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.
Variance = σ²
Toss of Three Dice
Toss of Four Dice
Probability Distribution Functions Binomial Poisson Gaussian or Normal (the famous bell curve)
Binomial Distribution Function
Poisson Distribution Function
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?
Lambda = 3 DiscreteCumulative xp(x) ∑p(x)
Poisson Distribution, Lambda = 3
Poisson Distribution, Lambda = 1.25
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)”.
Excel Function F(2) = NORMDIST(2, 0, 1, TRUE) = StdDev Mean = 0 StdDev of Data = 1 Cumulative = True
If NORMDIST() set to FALSE…
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.
False Alarm Probability
How Many Sigmas?
In Excel… K B = NORMINV((1-P FA )^(1/N),0,1) False Alarm Probability MeanStdDev
Computing Alarm Setpoint
Simplify and Divide by Time
Slight detour … 2-sided distribution
In Excel… Two sided distribution… …=2*(NORMDIST(x, 0, 1, TRUE) – 0.5)
Getting Back to Alarm Setpoint…
MDA-Driven Alarm Setpoint
“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
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.