Astroparticle Physics

384 A Mathematical AppendixPoisson distributionGaussian distributionPoisson:f(r,µ)= µr e −µ, r = 0, 1, 2,... , µ>0 ;r!mean: 〈r〉 =µ, variance: σ 2 = µ.Gaussian:f(x,µ,σ 2 ) = 1{ }σ √ 2π exp (x − µ)2−2σ 2 , σ > 0 ;mean: 〈x〉 =µ, variance: σ 2 .Landau distributionApproximation for the Landau distribution:L(λ) = √ 1 {exp − 1 }2π 2 (λ + e−λ ) ,where λ is the deviation from the most probable value.5. Errors and Error Propagationmean valuevariancestandard deviationindependent, uncorrelatedvariablesMean value of n independent measurements:〈x〉 = 1 nn∑x i ;i=1variance of n independent measurements:s 2 = 1 nn∑(x i −〈x i 〉) 2 = 1 ni=1n∑xi 2 −〈x〉2 ,i=1where s is called the standard deviation. A best estimate forthe standard estimation of the mean issσ = √ .n − 1If f(x,y,z) and σ x , σ y , σ z are the function and standarddeviations of the independent, uncorrelated variables, then( ) ∂f 2 ( ) ∂f 2 ( ) ∂f 2σf 2 = σx 2 ∂x+ σy 2 ∂y+ σz 2 ∂z.If D(z) is the distribution function of the variable z aroundthetruevaluez 0 with expectation value 〈z〉 and standarddeviation σ z , the quantity