1. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
Central limit theorem:
If is a sum of n independent randomly distributed variables with
arbitrary probability densities with expectation values <x> and a variances 2,
as , w becomes Gaussian with E[w] = n<x> and V[w] = n 2
a) Expectation:
b) Variance:
(remember V[x+y] = V[x] + V[y] + 2 cov[x,y])
K. Desch – Statistical methods of data analysis SS10

2. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
c) Take instead of xi
then:
The characteristic function of yi is:
For sum : →0
K. Desch – Statistical methods of data analysis SS10

3. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
Notes:
CLT depends crucially on convergence of the Taylor series
 integral of p.d.f.
 slow convergence in case of long tails
(Breit-Wigner distribution, Landau distribution)
generalization for sum of differently distributed p.d.f. mathematically different – convergence under certain conditions (generally met in
physical problems)
K. Desch – Statistical methods of data analysis SS10

4. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
Example:
energy loss per unit length of ionizing particles dE/dx
approximated by Landau distribution
No precise measurement of dE/dx
from single measurement i.e.
charge measurements in one detector layer
Many layers: approaches
Gaussian distribution
get rid of outliers by forming „truncated mean“ (reject the Fup (~70%)
largest single measurements)
K. Desch – Statistical methods of data analysis SS10

5. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
(~159 Messungen desselben Teilchens)
K. Desch – Statistical methods of data analysis SS10

6. K. Desch – Statistical methods of data analysis SS10
3. Distributions Central limit theorem
K. Desch – Statistical methods of data analysis SS10

7. K. Desch – Statistical methods of data analysis SS10
3. Distributions Uniform distribution
Example: digital readout of Si-strip detector
a b
pitch d=b-a
d = 50 μm
K. Desch – Statistical methods of data analysis SS10

8. K. Desch – Statistical methods of data analysis SS10
3. Distributions Uniform distribution
Each continuous probability density function f(x) with a known cumulative probability distribution F(x) could be transformed into a new variable, which will be distributed uniformly
Important part of Monte Carlo methods a
K. Desch – Statistical methods of data analysis SS10

9. K. Desch – Statistical methods of data analysis SS10
3. Distributions Breit-Wigner (Cauchy, Lorentz) distribution
: „Width“ = FWHM
Distribution has a long tail – not integrable
Mean, Variance, higher Moments not defined !
Physics: any resonance phenomenon
Sum BW-distributed variables:
BW-distribution
(CLT does not apply)
“Relativistic BW”:
K. Desch – Statistical methods of data analysis SS10

10. K. Desch – Statistical methods of data analysis SS10
3. Distributions 2 - distribution
x1,…,xn Gauss distributed random variables with i=0, i=1. Then the
Sum of squares, , follows to 2 – distribution with n degrees of
freedom
(x): Euler Gamma Function
(interpolation of factorial function)
2 – distribution plays an important role in statistical tests
fn(u) has a maximum by n-2
Mean E[u] = n
Variance V[u] = 2n
Approaches Gaussian for large n (CLT for xi2)
K. Desch – Statistical methods of data analysis SS10

11. K. Desch – Statistical methods of data analysis SS10
3. Distributions 2 - distribution
K. Desch – Statistical methods of data analysis SS10

12. K. Desch – Statistical methods of data analysis SS10
4. MC Methods Random generators
Monte Carlo methods are important statistical tools
Simulation of random processes
MC simulation is a method for iteratively evaluating a deterministic model using sets of random numbers as input (D.E. Knuth, ‘The art of computer programming’)
Random generators: quick, machine independent
Example: Linear congruential generator – pseudo-random algorithm
Ij = (a • Ij-1 + c ) mod m uj = Ij / m
a, c, m – constants; I0 is a ‘seed’
Period is at most m will be achieved when:
1. c ≠ 0
c and m are relatively prime
(a-1) is divisible by all prime factors of m
(a-1) is a multiple of 4 if m is a multiple of 4
Example: a = c = m =
K. Desch – Statistical methods of data analysis SS10

13. K. Desch – Statistical methods of data analysis SS10
4. MC Methods Random generators
‘Fibonacci generators’ based on generalization of Fibonacci sequence;
for example: Un = (Un-24 + Un-55) mod m
Need to be initialized !
Generators in root
TRandom P ≈ (linear congruential – don´t use!!)
TRandom1 P ≈ 10171
TRandom2 P ≈ 1026
TRandom3 P ≈
Tests for random generators
Uniform distribution: [0,1] ???
Correlation test:
Fill successive pairs in a 2D-histogram (see Blobel)
K. Desch – Statistical methods of data analysis SS10