Statistical Methods in Particle Physics / WS 13 Lecture II Probability Density Functions Niklaus Berger Physics Institute, University of Heidelberg
Recap of Lecture I: Kolmogorov Axioms Ingredients: Set S of outcomes of the experiment (sample space) Subsets A, B... of S (technically need to form a Σ-Algebra) Mapping into the real numbers P(A) called the Probability Kolmogorov Axioms: For all A S, P(A) 0 P(S) = 1 If A B = 0, P(A B) = P(A) + P(B) Niklaus Berger SMIPP WS 2013 Slide 2
Recap of Lecture I: Bayes Theorem Niklaus Berger SMIPP WS 2013 Slide 3
Part II: Catalogue of Probability Density Functions Niklaus Berger SMIPP WS 2013 Slide 4
2.1. Binomial Distribution 0.00 0.05 0.10 0.15 0.20 0.25 p=0.5 and n=20 p=0.7 and n=20 p=0.5 and n=40 P (x; n, p) = (x) n px (1-p) n-x B Mean: E[x] = np 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 p=0.5 and N=20 p=0.7 and N=20 p=0.5 and N=40 0 10 20 30 40 Variance: V[x] = np(1-p) Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 5
Binomial Distribution: Application y z Efficiency measurement: x n particles cross the Device Under Test (DUT) DUT has an efficiency to detect a particle of p Average number of hits seen in DUT x is n p Results of many experiments will follow a binomial distribution - use corresponding error bars Do NOT use counting errors on n and x and perform error propagation Niklaus Berger SMIPP WS 2013 Slide 6
Rare events Take limit of binomial distribution for n p 0 with n p fixed to ν Rare events with constant rate Niklaus Berger SMIPP WS 2013 Slide 7
2.2. Poisson Distribution ν x P P (x; ν) = x! e -ν Mean: Variance: E[x] = ν V[x] = ν Standard deviation: Σ = ν Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 8
Poisson Distribution: Application Any counting experiment with rare events will have results that are Poisson distributed: ν = Σ Ldt expected number of events given luminosity and cross-section Number of entries in a histogram bin Number of cosmic rays passing through you per second (what is ν?) Niklaus Berger SMIPP WS 2013 Slide 9
Many rare events Look at Poisson distribution for ν >> 1 Many Rare events with constant rate (and almost every other large N limit...) Niklaus Berger SMIPP WS 2013 Slide 10
2.3. Normal (Gaussian) Distribution 1 P (x; μ,σ) = e -(x-μ)2 /2Σ 2 G 2πΣ 1.0 0.8 μ= 0, μ= 0, μ= 0, μ= 2, 2 σ = 0.2, 2 σ = 1.0, 2 σ = 5.0, 2 σ = 0.5, φ μ,σ 2( x) 0.6 0.4 0.2 Mean: E[x] = μ 0.0 1.0 0.8 5 4 3 2 1 0 1 2 3 4 5 μ= 0, μ= 0, μ= 0, μ= 2, 2 σ = 0.2, 2 σ = 1.0, 2 σ = 5.0, 2 σ = 0.5, x Variance: V[x] = Σ 2 Φ μ,σ 2(x) 0.6 0.4 CDF is called error function 0.2 0.0 5 4 3 2 1 0 1 2 3 4 5 x Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 11
Central limit theorem The arithmetic mean of a sufficiently large number of uncorrelated random variables drawn from a distribution with well defined mean and well defined and finite variance will be approximately normal distributed This is independent of the underlying distribution If your measurement is sufficiently messy (i.e. influenced by enough independent effects) it will be normal distributed Usually leads to the assumption, that everything is normal distributed, which is wrong... Niklaus Berger SMIPP WS 2013 Slide 12
Examples for the normal distribution Good example: Size distribution among students of a particular sex in a class Not so good example: Distribution of scattering angles of particles passing through a slab of material (There are large angle scatters (Rutherford!) that produce non-gaussian tails) Bad example: Energy loss of particles passing through a slab of material (Dominated by few large losses - Landau distribution) Niklaus Berger SMIPP WS 2013 Slide 13
Quantiles of the Normal Distribution: Standard Deviations 0.0 0.1 0.2 0.3 0.4 0.1% 2.1% 3σ 34.1% 34.1% 13.6% 13.6% 2.1% 0.1% 2σ 1σ µ 1σ 2σ 3σ Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 14
2.4. Log Normal Distribution 1 P (x; μ,σ) = e -(ln x-μ)2 /2Σ 2 G x 2πΣ Mean: E[x] = e μ + Σ2 /2 Variance: V[x] = (e Σ2-1) e 2μ+Σ2 Order of magnitude is normally distributed: e.g. growth of droplets Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 15
2.5. Exponential Distribution 1 P (x; τ) = e - x/τ for x 0 E Τ Mean: E[x] = Τ Variance: V[x] = Τ 2 Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 16
Exponential Distribution: Application Lifetime of a particle Has no memory: f(t -t 0 t > t 0 ) = f(t) Niklaus Berger SMIPP WS 2013 Slide 17
2.6. Uniform Distribution 1 P (x; a, b) = for a x b U a-b Mean: E[x] = 1/2 (a+b) Variance: V[x] = 1/12 (a-b) 2 Plot source: Wikipedia Niklaus Berger SMIPP WS 2013 Slide 18