Download Lecture 3: Probability Distributions in Statistical Data Analysis by G. Cowan and more Slides Computational and Statistical Data Analysis in PDF only on Docsity!
Statistical Data Analysis: Lecture 3
1 Probability, Bayes’ theorem, random variables, pdfs 2 Functions of r.v.s, expectation values, error propagation 3 Catalogue of pdfs 4 The Monte Carlo method 5 Statistical tests: general concepts 6 Test statistics, multivariate methods 7 Goodness-of-fit tests 8 Parameter estimation, maximum likelihood 9 More maximum likelihood 10 Method of least squares 11 Interval estimation, setting limits 12 Nuisance parameters, systematic uncertainties 13 Examples of Bayesian approach 14 tba 15 tba
Some distributions
Distribution/pdf Example use in HEP Binomial Branching ratio Multinomial Histogram with fixed N Poisson Number of events found Uniform Monte Carlo method Exponential Decay time Gaussian Measurement error Chi-square Goodness-of-fit Cauchy Mass of resonance Landau Ionization energy loss Beta Prior pdf for efficiency Gamma Sum of exponential variables Student’s t Resolution function with adjustable tails
Binomial distribution (2)
The binomial distribution is therefore random variable parameters For the expectation value and variance we find:
Binomial distribution (3)
Binomial distribution for several values of the parameters: Example: observe N decays of W ± , the number n of which are W→μν is a binomial r.v., p = branching ratio.
Multinomial distribution (2)
Now consider outcome i as ‘success’, all others as ‘failure’. → all n i individually binomial with parameters N , p i for all i One can also find the covariance to be Example: represents a histogram with m bins, N total entries, all entries independent.
Poisson distribution
Consider binomial n in the limit → n follows the Poisson distribution: Example: number of scattering events
n with cross section σ found for a fixed
integrated luminosity, with
Exponential distribution
The exponential pdf for the continuous r.v. x is defined by: Example: proper decay time t of an unstable particle
( τ = mean lifetime)
Lack of memory (unique to exponential):
Gaussian distribution
The Gaussian (normal) pdf for a continuous r.v. x is defined by:
Special case: μ = 0, σ
2 = 1 (‘standard Gaussian’):
(N.B. often μ, σ
2 denote mean, variance of any r.v., not only Gaussian.)
If y ~ Gaussian with μ, σ
2
, then x = ( y μ) / σ follows ϕ ( x ).
Central Limit Theorem (2)
The CLT can be proved using characteristic functions (Fourier transforms), see, e.g., SDA Chapter 10. Good example: velocity component v x of air molecules. OK example: total deflection due to multiple Coulomb scattering. (Rare large angle deflections give non-Gaussian tail.) Bad example: energy loss of charged particle traversing thin gas layer. (Rare collisions make up large fraction of energy loss, cf. Landau pdf.) For finite n , the theorem is approximately valid to the extent that the fluctuation of the sum is not dominated by one (or few) terms. Beware of measurement errors with non-Gaussian tails.
Multivariate Gaussian distribution
Multivariate Gaussian pdf for the vector are column vectors, are transpose (row) vectors, For n = 2 this is
where ρ = cov[ x
1 , x 2
]/( σ
1
2 ) is the correlation coefficient.
Cauchy (Breit-Wigner) distribution
The Breit-Wigner pdf for the continuous r.v. x is defined by (Γ = 2, x 0 = 0 is the Cauchy pdf.) E [ x ] not well defined, V [ x ] →∞. x 0 = mode (most probable value) Γ = full width at half maximum Example: mass of resonance particle, e.g. ρ, K
, φ 0 , ... Γ = decay rate (inverse of mean lifetime)
Landau distribution
For a charged particle with β = v / c traversing a layer of matter
of thickness d , the energy loss Δ follows the Landau pdf:
L. Landau, J. Phys. USSR 8 (1944) 201; see also W. Allison and J. Cobb, Ann. Rev. Nucl. Part. Sci. 30 (1980) 253.
d
Beta distribution
Often used to represent pdf of continuous r.v. nonzero only between finite limits.
Gamma distribution
Often used to represent pdf of continuous r.v. nonzero only in [0,∞]. Also e.g. sum of n exponential r.v.s or time until n th event in Poisson process ~ Gamma
