Statistics for Particle Physics Lecture 1: Distributions

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Statistics for Particle PhysicsLecture 1
Distributions

• Roger Barlow
• Manchester University
• TAE, Santander,10 July 2006

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Statistics Why?

• Statistics is not Physics. It is not
  fundamental.
• But it is essential in our understanding of the
  experimental results which do led us towards the
  fundamental truths.
• We learn this the hard way, as individuals and as
  a community

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Two sorts of Statistics

• Descriptive Statistics
• How to convey information from complex data to
  audience (public, colleagues, superiors)
• Simple numbers and plots
• Driven by convention and common sense
• Used by businessmen, journalists, and almost
  everybody
• An Art

• Inferential Statistics
• What the data is trying to tell us about the
  process that produced it
• Algorithms may be simple or complicated
• Rigorous and verifiable
• Used by Physicists, doctors, and farmers
• A Science

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Topic Distributions

• Any non-deterministic model involves random
  distributions. (Discrete or continuous)
• Lots of these Uniform, exponential, Chi-squared,
  
• 3 most important
• Binomial
• Poisson
• Gaussian

r
x
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Binomial P(rn,p)

• Tossing n coins. Probability of r heads and n-r
  tails is
• P(rp,n)__n!__ pr(1-p) n-r
• r! (n-r)!
• Mean is ?np. Width described by ??np(1-p)
• 1-p is often written q. Makes the formulae more
  elegant

5 trials
50 probability
90 probability
10 probability
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When do we meet it?

• Numbers of hits in Tracking chambers
• Track-finding and Particle ID event efficiencies
  and multiplicities
• Monte Carlo event generation (but see later)
• Honestly, not very often

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Poisson

• Probability of r events when the mean is ? is
  P(r?)e- ? ? r
• r!
• Mean is ?. Standard deviation is ??.
• Works for large and small ?

Mean 5.4
Mean 0.2
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When do we get a Poisson distribution?

• Discrete events in a continuum
• Radioactive decays Geiger counter clicks are
  classic example
• Numbers of events in data in particular numbers
  of rare events found. And numbers of events in
  histogram bins

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The Gaussian
1 . ??2?

• P(x?,?) exp (-(x- ?)2/ 2?2)

There is only one universal Gaussian picture.
Everything else is scaling and translation.
Peaks at x ? Width described by ? Bell-shaped
curve
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When do we get Gaussians

• Lots! Everything becomes Gaussian in some large
  n limit thanks to the Central Limit Theorem (CLT)
• If y is a random variable made by the
  convolution of n random variables xi, then
• my?mxi
• ?2y??2xi
• The distribution function for y becomes Gaussian
  for large n

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The Standard Deviation

• Any distribution has a mean ??x P(x) dx and a
  standard deviation given by
• ?2?(x-?)2 P(x) dx
• Standard deviations add in quadrature (apart from
  complications due to correlations.)
• For the Gaussian only, 68 of P(x) lies with ? ?,
  95 within ? 2 ?, 99.7 within ? 3 ?

Apart from some pathological exceptions you
should pray you never meet
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Convolutions that help us

• Poisson Poisson Poisson
• Separate sources add and can be treated as a
  single source
• Poisson Binomial Poisson
• A poisson source modified by a binomial detection
  efficiency gives a poisson number of detected
  events

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Limit Binomial ? Poisson

• Fixing n?p as n increases binomial becomes
  Poisson
• So we can use Poisson for MC data, which is truly
  binomial (unlike real data, which is Poisson)

Poisson(r,0.2)
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Limit Binomial ? Gaussian
All with p0.2
N10
N20
N80
N40
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Limit Poisson ? Gaussian
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Now try and example

• Gaussian uniform