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Monte Carlo Methods for Evaluating Complex Integrals and Probability Distributions

This document outlines various Monte Carlo methods utilized for evaluating challenging integrals, including cross sections and decay rates. It explains the fundamentals of sampling complex distribution functions, such as uniform and exponential distributions. The examples provided illustrate the use of ROOT's TRandom for random number generation and demonstrate techniques like the acceptance-rejection method and importance sampling. Additionally, it discusses the practical aspects of implementing these methods in ROOT, a widely used toolkit in particle physics.

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Monte Carlo Methods for Evaluating Complex Integrals and Probability Distributions

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  5. CMSSW ROOT P.Miné Monte Carlo

  6. Monte-Carlo methods are used to: - evaluate difficult integrals examples : cross section, decay rate - sample complicated distribution function The simplest pdf (probablity distribution function): uniform 0 < x < 1 all values of x have the same probabilty f(x) =1 Programs exist to give a value of x at each call: Example: TRandom in ROOT P.Miné Monte Carlo

  7. Random generators simple cases Exponential distribution (decay time of a particle) f(t) = (1 / τ ) exp(- t / τ ) in the interval [a, b] let a’ = exp(- a / τ ) and b’ = exp(- b / τ ) t = - τ ln ( b’ + u ( a’–b’) ) where u is uniform in [0, 1] then dN / dt = (dN / du) (du / dt) = 1 x (1 / τ ) exp(- t / τ ) / (a’ –b’) For [0 , ∞] take simply t = - τ ln u This method works every time the primitive F(x) of f(x) is invertible x = F-1(u) where u is a uniform probability variable P.Miné Monte Carlo

  8. Isotropy in 3D Density is proportional to solid angle dΩ = d(cosθ) dϕ cosθ is uniform in [-1, 1] : take 2u1 -1 ϕ uniform in [0, 2π] Gaussian variable are gaussian independent P.Miné Monte Carlo

  9. The acceptance rejection method (Von Neumann) • Assume that for any x, the probability distribution function f(x) can be calculated and its maximum C is known Take the envelope Ch(x) ,h(x) is uniform f(x) and h(x) are normalized so C > 1 For each value of x, generate the random variable u, with uniform distribution If u C h (x) < f(x) , accept x ; if not , reject Try again P.Miné Monte Carlo

  10. Importance sampling Increase the efficiency if C >> 1 by changing h(x) The best solution: change the variable (Jacobian method) P.Miné Monte Carlo

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  16. ν l W t b P.Miné Monte Carlo

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  19. Proton proton cross section P.Miné Monte Carlo

  20. Parton density functions are obtained experimentally by deep inelastic scattering of leptons P.Miné Monte Carlo

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  22. Practiceon ROOT Dowload from root.cern.ch/root/root_v5.22.00.source.tar.gz gzip -dc root-v5.22.00.source.tar.gz | tar –xf export ROOTSYS=<path>/root ./configure –help ./configure [<arch>] [set arch appropriately if not default] (g)make export PATH=$ROOTSYS/bin:$PATH export LD_LIBRARY_PATH=$ROOTSYS/lib:$LD_LIBRARY_PATH Exercise in tutorials P.Miné Monte Carlo

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