Probability Distributions: Applications

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Probability Distributions Applications

• Applications of Gaussian distribution
• The standard error on the mean
• Reduced Gaussian for uncertainty estimates
• Use of Poisson for counting statistics
• Null experiments (confidence)

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Examples of Applications

• We need to find the best estimate of the object
  length.
• The errors in this measurement are instrumental ?
  the errors are Gaussian. The mean value is
• The standard deviation is
• The standard error in the mean
• The best value of the length (17.6150.002)m

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Special case Poisson

• In counting experiments uncertainties are best
  described by Poisson statistics.
• Statistical fluctuations are independent of time
  the resulting errors are called statistical
  errors.
• The best estimate of mean per unit time in
  Poisson statistics
• Variance of the sample mean, , is given by
  the variance of the parent distribution, s,
  divided by the sample size, N

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Special case Poisson

• Remember for the Poisson distribution, the
  variance is equal to the mean, the variance of
  the sample mean is therefore,
• And the standard deviation of the mean is,
• Where m is the average per unit time.

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Poisson Example

• Determination of count rates and their errors
• The follow measurements of counts/min were made
  from a GM tube viewing a 22Na source.
  2201,2145,2222,2160,2300
• What is the decay rate?
• Using Poisson estimators
• The standard error on the mean
• The best estimate of the decay rate
  (220621)counts/min

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Poisson Example

• What would have happened if we had instead
  counted for one five minute period?
• 11028 counts. Constitutes sample N1
• Mean count rate for 5 minutes is 11028.
• The error is
• Number of counts/min 220621, identical to
  before.
• A common error to avoid is to calculate the
  counts per minute and then take the square root
  of this number.

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Poisson Examples

• Phone calls are received at Mandys residence as
  a Poisson process with parameter l2 per hour?
• If Mandy takes a 10 minute shower, what is the
  probability that the phone rings during that
  time?
• Probability of a ring (or more than once) 0.284

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Gaussian Further examples

• If we measure the speed of sound in an experiment
  to be 38020m/s. We would like to know how
  consistent our measurement is with a theory which
  predicts it to be 340m/s.
• We need to consider the fractional area under the
  tails of the Gaussian distribution, or the area
  with t greater that some specified value of r,
  where t is the distance from the mean, measured
  in standard deviation units.
• You can determine tgtr or tgtr, the former gives
  the area in the tail of one side of the symmetric
  distribution. This is useful when you have
  xDx1and x-Dx2. At r0 the probability or area of
  one tail is just half that of two tails.

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Factional area under tail
t0
-r
r
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Integrated Gaussian distribution
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Fractional area under tail

• In this example
• The corresponding probability is 4.6 (from
  table).
• If 1000 experiments were performed, we would
  expect 46 of them would differ from the predicted
  value by at least as much as ours.

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Confidence limits and Null Experiments

• Setting confidence limits when no counts are
  observed
• Many physics experiments test the validity of
  certain theoretical conservation laws, by the
  search for reactions of decays forbidden by these
  laws.
• If a observation is made for a certain amount of
  time, if one of more events are observed the
  theoretical law is disproved.
• If NO events are observed the converse is NOT
  true.
• Instead the limit on the lifetime of the reaction
  or decay is set.

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Null experiments

• If the process has some mean reaction rate, l,
  then the probability of observing no counts in
  the time period, T, is
• Remember, P(0l) can be interpreted as the
  probability distribution for l when no counts are
  observed in a period T.
• Question What is the probability that l is less
  than some value lo?

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Null experiments

• Question What is the probability that l is less
  than some value lo?
• This probability is known as the confidence level
  for the interval between 0 and l0. To make a
  strong statement we chose a high confidence level
  (CL) for example 90 (CL0.9).

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Null experiments Example

• 50g of 82Se is observed for 100 days, to look for
  a neutrino-less double beta decay a reaction
  normally forbidden by lepton conversion. However
  theory suggests this process may occur.
• The apparatus has a detection efficiency of 20.
  No events were recorded.
• For 90 confidence
• For 50g sample, the total number of nuclei is
• Limit on decay rate/nucleus is
• The lifetime is just the inverse of l

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Summary of lecture 4

• Applications of Gaussian distribution
• The standard error on the mean
• Reduced Gaussian for uncertainty estimates
• Use of Poisson for counting statistics
• Null experiments (confidence)

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Summary of Excel Probability Functions

• Binomial
• BINOMDIST(r,N,p,false)
• Poisson
• POSSON(x,m,false)
• Gaussian
• NORMDIST(x,m,s,false)

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Gaussian probability