Uncertainty, Probability, and Statistics Part 3 Statistics

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Uncertainty, Probability, and StatisticsPart 3
- Statistics
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The probability of an event can be expressed as a
binomial probability if its outcomes can be
broken down into two probabilities p and q, where
p and q are complementary (i.e. p q 1) For
example, tossing a coin can be either heads or
tails, each which have a (theoretical)
probability of 0.5. Rolling a four on a six-sided
die can be expressed as the probability (1/6) of
getting a 4 or the probability (5/6) of rolling
something else.The Poisson distribution is a
discrete probability distribution that expresses
the probability of a number of events occurring
in a fixed period of time if these events occur
with a known average rate and independently of
the time since the last event. The Poisson
distribution can also be used for the number of
events in other specified intervals such as
distance, area or volume. Nuclear counting
statistics!The Gaussian distribution is a
continuous probability distribution, applicable
in many fields. It may be defined by two
parameters the mean ("average", µ) and variance
(standard deviation squared) s2, respectively.
The importance of the normal distribution as a
model of quantitative phenomena in the natural
sciences is due to the central limit theorem.
Many physical phenomena (like noise) can be
approximated well by the normal distribution.
While the mechanisms underlying these phenomena
are often unknown, the use of the normal model
can be theoretically justified by assuming that
many small, independent effects are additively
contributing to each observation.
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The Binomial DistributionElementary example
Roll a die ten times and count the number of
sixes. The distribution is a binomial one with n
10 and p 1/6.

• n trials r successes
• Individual success probability p

Variance V???lt(r- ? )2gtltr2gt-ltrgt2 np(1-p)
Mean ?ltrgt?rP( r ) np

• Met with in Efficiency/Acceptance calculations

1-p ??p ? q
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Poisson

• ? is a positive real number, equal to the
  expected number of occurrences during a given
  interval. For instance, if the events occur on
  average every 4 minutes, and you are interested
  in the number of events occurring in a 10 minute
  interval, you would use a Poisson distribution
  with ? 10/4 2.5.

Mean ?ltrgt?rP( r ) ?
Variance V???lt(r- ? )2gtltr2gt-ltrgt2 ?
For this distribution, the probability of getting
a certain result r when the true value is ??is
given by P
NOTE The function is only non-zero at positive
integer values of r. The connecting lines are
only guides for the eye and do not indicate
continuity.
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Binomial and PoissonPoisson - If a mean or
average probability of an event happening per
unit time/per page/per mile, etc., is given, and
you are asked to calculate a probability of n
events happening in a given time/number of
miles/etc.Binomial - If an exact probability of
an event happening is given, or implied, and you
are asked to calculate the probability of this
event happening r times out of n, then the
Binomial Distribution must be used.

• A student is standing by the road, hoping to
  hitch a lift. Cars pass according to a Poisson
  distribution with a mean frequency of 1 per
  minute. The probability of an individual car
  giving a lift is 1. Calculate the probability
  that the student is still waiting for a lift
• (a) After 60 cars have passed
• (b) After 1 hour

• 0.99600.5472

b) e-0.60.5488
Check out http//www.graphpad.com/quickcalcs/proba
bility1.cfm
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Stopped Here

• Lets review from last time.
• Talked about uncertainties in measurement
• Blunders
• Systematic effects / Systematic uncertainties
• Theoretical uncertainties
• Random / statistical uncertainties
• For the latter, can quantfy if we know the
  probability distribution
• Governing ideas of probability distributions

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Moved to the practical -
Looked at Binomial, Poisson distributions..
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The Binomial DistributionElementary example
Roll a die ten times and count the number of
sixes. The distribution is a binomial one with n
10 and p 1/6.

• n trials r successes
• Individual success probability p

Variance V???lt(r- ? )2gtltr2gt-ltrgt2 np(1-p)
Mean ?ltrgt?rP( r ) np

• Met with in Efficiency/Acceptance calculations

1-p ??p ? q
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Poisson

• ? is a positive real number, equal to the
  expected number of occurrences during a given
  interval. For instance, if the events occur on
  average every 4 minutes, and you are interested
  in the number of events occurring in a 10 minute
  interval, you would use a Poisson distribution
  with ? 10/4 2.5.

Mean ?ltrgt?rP( r ) ?
Variance V???lt(r- ? )2gtltr2gt-ltrgt2 ?
For this distribution, the probability of getting
a certain result r when the true value is ??is
given by P
NOTE The function is only non-zero at positive
integer values of r. The connecting lines are
only guides for the eye and do not indicate
continuity.
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Binomial and PoissonPoisson - If a mean or
average probability of an event happening per
unit time/per page/per mile, etc., is given, and
you are asked to calculate a probability of n
events happening in a given time/number of
miles/etc.Binomial - If an exact probability of
an event happening is given, or implied, and you
are asked to calculate the probability of this
event happening r times out of n, then the
Binomial Distribution must be used.

• A student is standing by the road, hoping to
  hitch a lift. Cars pass according to a Poisson
  distribution with a mean frequency of 1 per
  minute. The probability of an individual car
  giving a lift is 1. Calculate the probability
  that the student is still waiting for a lift
• (a) After 60 cars have passed
• (b) After 1 hour

• 0.99600.5472

b) e-0.60.5488
Check out http//www.graphpad.com/quickcalcs/proba
bility1.cfm
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Binomial tends to Poisson if
individual probability - from binomial
Example ??5
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Variance

• The probability that a given measurement will be
  close to the true value depends on the relative
  width of the frequency curve.
• This is quantified by the variance, s2, of the
  distribution.
• Related to statistical index called standard
  deviation, measure of dispersion of measurement
  results about the mean
• The variance is a number such that 2/3 of the
  measurement falls within /- 1s (standard
  deviation) of the true value.
• For Poisson, s2 ??(true value)

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See board - work an example
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Gaussian
f(x)

• The continuous probability density function of
  the normal distribution is the Gaussian function
  

standard normal distribution
x
Mean ?ltxgt?xP( x ) dx ?
Variance V???lt(x- ?)2gtltx2gt-ltxgt2 ??
http//davidmlane.com/hyperstat/normal_distributio
n.html
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Probability Contents

• 68.27 within 1?
• 95.45 within 2?
• 99.73 within 3?

90 within 1.645 ? 95 within 1.960 ? 99 within
2.576 ? 99.9 within 3.290?
These numbers apply to Gaussians and only
Gaussians
Other distributions have equivalent values which
you could use if you wanted
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• also known as Breit-Wigner - resonance mass
  distribution!
• (2) x time to observe n events in a Poisson
  process

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Back to the Gaussian - why?

• One remarkable fact about the normal distribution
  is the fact that if we took many samples of size
  n from a population having mean ??and variance ?2
  (that is, any distribution we want), then -
• the population of xs would be approximately
  normally distributed with mean ? and variance
  ?2/n.
• The larger n is, the better the approximation is.
• These facts are known collectively as the Central
  Limit Theorem and allow us to make inferences
  about population means using the normal
  distribution no matter what the distribution of
  the population being sampled from.

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CLT in real life

• Examples
• Height
• Simple Measurements
• Student final marks

• Counterexamples
• Weight
• Wealth

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Measurement Uncertainty- Why This All Matters!

• Have a single measurement N
• Dont know exact value of ??or ?
• Can reasonably assume N ?
• Therefore?????sqrt(N). Recall, for Poisson, s2
  ??(true value).
• So - if the result of a measurement is N, there
  is a 68.3 chance that the true value of the
  measurement is within the range N /- sqrt(N)
• This is the 1 ??confidence interval - there are
  others
• Sqrt(N) is the uncertainty on N (use percentage,
  100/sqrt(N)
• Compare the percentage uncertainties in the
  measurements N1 100 counts and N2 10,000
  counts

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Background Events - an example in how to
propogate errors corectly!

• For a series of measurements M1, M2, M3, the
  individual variances will be ?(M2)2, ?(M2)2,
  ?(M3)2,
• The general equation for the variance of the
  result is given by ?(M1 M2 M3 )
  sqrt?(M2)2?(M2)2?(M3)2.
• So, for a given series of counting measurements
  with individual results N1, N2, N3,, the
  variance of the result is given by.?..

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Background Events - an example in how to
propogate errors corectly!

• For a series of measurements M1, M2, M3, the
  individual variances will be.?.?(M2)2, ?(M2)2,
  ?(M3)2,
• The general equation for the variance of the
  result is given by ?(M1 M2 M3 )
  sqrt?(M2)2?(M2)2?(M3)2.
• So, for a given series of counting measurements
  with individual results N1, N2, N3,, the
  variance of the result is given by.?..sqrtN1
  N2 N3 . and the precentage uncertainty is
  sqrtN1 N2 N3 . / (N1 N2 N3 ) x
  100.
• If background present counting rate Rs(ample)
  Rg(ross) - Rb(ackground)
• Uncertainty in ??Rs) sqrtRg/tg Rb/tg, where
  t counting time

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Put numbers in.

• In 4 minute counting measurements, gross sample
  counts are 6000 counts and background counts are
  4000 counts. What are the sample and gross
  counting rates and uncertainties? What would the
  sample rate and uncertainty be if there were no
  background?

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• ?Rs 25 cpm
• ?Rg 19 cpm (1)
• ?RsNB 11 cpm (2)
• Conclusions
• High background rates undesirable! Increase
  uncertainties in net counting rates
• Small differences between relatively high
  counting rates can have relatively large
  uncertainties

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Fitting and Chi squared - switch to .pdf document!

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Also look at t-tests

• Less common, but you will see occasionally and
• there will be homework