Statistical Methods in Particle Physics

Transcription

1 Statistical Methods in Particle Physics. Probability Distributions Prof. Dr. Klaus Reygers (lectures) Dr. Sebastian Neubert (tutorials) Heidelberg University WS 07/8

2 Gaussian g(x; µ, )= p exp (x µ) Mean: E[x] =µ Variance: V [x] = μ = 0, σ = ("standard normal distribution"): (x) = p e x Cumulative distribution related to error function: (x) = p Z x e z dz = apple erf xp + Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions

3 p-value Probability for a Gaussian distribution corresponding to [μ Zσ, μ +Zσ]: P(Z )= p Z +Z Z e x dx = (Z) ( Z) =erf Zp 68.7% of area within ±σ 95.45% of area within ±σ 99.73% of area within ±3σ p-value: probability that a random process produces a measurement thus far, or further, from the true mean p-value = P(Z ) In root: TMath::Prob standard to report a discovery 90% of area within ±.645σ 95% of area within ±.960σ 99% of area within ±.576σ Two-sided Gaussian p-values Deviation p-value (%) σ 3.7 σ σ σ σ Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 3

4 Why Are Gaussians so Useful? Central limit theorem: When independent random variables are added, their properly normalized sum tends toward a normal distribution (a bell curve) even if the original variables themselves are not normally distributed. More specifically: Consider n random variables with finite variance σi and arbitrary pdf: y = nx n! nx x i E[y] = µ i V [y] = i= Measurement uncertainties are often the sum of many independent contributions. The underlying pdf for a measurement can therefore be assumed to be a Gaussian. Many convenient features in addition, e.g., sum or difference of two Gaussian random variables is again a Gaussian. i= nx i= i Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 4

5 The CLT at Work A: x taken from a uniform PD in [0,], with µ=0.5 and σ =/, N=5000 B: X = x +x from A, N=5000, flat shoulders C: X = x +x +x 3 from A, curved shoulders D: X=x +x + +x from A, almost Gaussian Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 5

6 Multivariate Gaussian transposed (row) vectors column vectors f (~x; ~µ, V )= exp ( ) N/ V / apple (~x ~µ)t V (~x ~µ) ~x =(x,...,x n ), ~µ =(µ,...,µ n ) E[x i ]=µ i V i,j = cov[x i, x j ]=h(x i µ i )(x j µ j )i For n = : V = x x y x y y V = ( ) / x /( x y ) /( x y ) / y ρ = correlation coefficient Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 6

7 d Gaussian Distribution and Error Ellipse We obtain the d Gaussian distribution: f (x, x ; µ, µ,,, ) = p " x µ x µ exp ( + ) x µ x µ #! where ρ = cov(x, x)/(σσ) is the correlation coefficient. Lines of constant probability correspond to constant argument of exp this defines an ellipse σ ellipse: f(x, x) has dropped to / e of its maximum value (argument of exp is /): x µ + x µ x µ x µ = Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 7

8 d Gaussian: Error Ellipse Ellipse which contains 68% of the events f y (x) = = Z f (x, y)dy p x exp x µx x! σ ellipse (/ e of maximum values) f x (y) = p y exp y µy y! Physics 509 P D P D σ σ σ σ σ σ Probability for an event to be within σ ellipse: 39.34% Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 8

9 Poisson Distribution p(k; µ) = µk k! e µ μ = E[k] =µ, V [k] =µ μ = 4 μ = 0 Properties: n, n follow Poisson distr. n+n follows Poisson distr., too Can be approximated by a Gaussian for large ν Examples: Clicks of a Geiger counter in a given time interval Number of Prussian cavalrymen killed by horse-kicks Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 9

10 Binomial Distribution N independent experiments Outcome of each is 'success' or 'failure' Probability for success is p f (k; N, p) = N k = N! k!(n N p k ( p) N k E[k] =Np V [k] =Np( p) k k)! binomial coefficient: number of different ways (permutations) to have k successes in N tries Use binomial distribution to model processes with two outcomes Example: Detection efficiency (either we detect particle or not) For small p, the binomial distribution can be approximated by a Poisson distribution (more exactly, in the limit N, p 0, N p constant) Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 0

11 Negative Binomial Distribution Keep number of successes k fixed and ask for the probability of m failures before having k successes: m + k E[m] =k P(m; k, p) = p p k ( p) m p m m = 0,,..., V [m] =k p Another representation: m + k P(m; µ, k) = m µ k + µ k m m+k E[m] =µ V [m] =µ p + µ k Use Gamma-fct. for non-integer values x! := (x + ) p = + µ k [relation btw. parameters] Example: Distribution of the number of produced particles in e + e and proton-proton collisions reasonably well described by a NBD. Why? Empirical observation, not so obvious. Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions

12 Uniform Distribution f (x; a, b) = Properties: ( b a, a apple x apple b 0, otherwise b a E[x] = (a + b) 0 a b V [x] = (b a) Example: Strip detector: resolution for one-strip clusters: pitch/ Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions

13 Exponential Distribution f (x; ) = ( e x/ x 0 0 otherwise E[x] = V [x] = Example: Decay time of an unstable particle at rest f (t, ) = e t/ = mean lifetime Lack of memory (unique to exponential): f (t > t 0 + t t > t 0 )=f (t > t ) Probability for an unstable nucleus to decay in the next minute is independent of whether the nucleus was just created or already existed for a million years Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 3

14 Landau Distribution Z L. Landau, J. Phys. USSR 8 (944) 0 W. Allison and J. Cobb, Ann. Rev. Nucl. Part. Sci. 30 (980) 53. Describes energy loss of a charged particle in a thin layer of material tail with large energy loss due to occasional creation of delta rays f ( )= Z e u ln u 0 u sin( u)du 5 f (λ) = ϖ -u ln u - λu e sin (ϖu) du 0 actual energy loss location parameters f (λ) 0 = 0 material property λ Unpleasant mathematical properties: mean and variance not defined root: TMath::Landau() Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 4

15 [Delta rays] Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 5

16 Student's t Distribution Let x,, xn be distributed as N(μ, σ). Developed in 908 by William Gosset for the Guinness Brewery. Published under the name "student". Sample mean and estimate of the variance: x = n nx i= x i ˆ = n nx (X i X ) i= How Student's distribution arises from sampling: x µ / p n follows standard normal distr. (μ=0, σ=) x µ ˆ/ p n not Gaussian. Student's t distr. with n degrees of freedom Student's t distribution: f (t; n) = ( n+ ) p n ( n ) + t n n+ n =: Cauchy distr. n!: Gaussian Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 6

17 χ Distribution Let x,, xn be n independent standard normal (μ = 0, σ = ) random variables. Then the sum of their squares nx z = follows a χ distribution with n degrees of freedom. χ distribution: i= x i f (z; n) = z (n/ ) e z/ n/ n E[z] =n, V [z] =n (z 0) Application: Quantifies goodness of fit nx yi g(x i ) = i i= Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 7

18 Log-Normal Distribution Let y be a normal (i.e. Gaussian) distributed random variable. Then x = exp(y) follows the log-normal distribution f (x; µ, )= x (ln x µ) p exp E[x] =exp µ + V [x] =[exp( ) ] exp(µ + ) Multiplicative version of the central limit theorem Relevant when observable is product of fluctuating variables Occurs frequently, e.g., city sizes Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 8

19 Cauchy, Breit-Wigner, or Lorentzian Distribution Particle physics: cross section for production of resonance with mass M and width Γ (full width at half maximum): f (E; M, )= (E M) +( /) Dimensionless form: f (x) = +x x = E M / here: x0 = M, x = E Mean and variance are undefined, mode is M. Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 9

20 Cumulative Distribution Function F (X ):= Z x f (x 0 )dx 0 Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions 0

21 Convolution of Probability Distributions f(x): probability distribution of random variable x g(y): probability distribution of random variable y PDF for sum is given by: z = x + y h(z) =(f g)(z) = Z f (z t)g(t)dt = Z f (t)g(z t)dt Example: Two Gaussians N(x; μx, σx), N(y; μy, σy) Sum z = x + y follows a Gaussian with µ = µ x + µ y, = q x + y Note: Product x y and ratio of x/y of two Gaussian distributed random variables is not a Gaussian Statistical Methods in Particle Physics WS 07/8 K. Reygers. Probability Distributions