1. A Monte Carlo exploration of methods to determine the UHECR composition with the Pierre Auger Observatory D.D’Urso for the Pierre Auger Collaboration http://www.auger.org/auger-authors-ICRC2009.pdf HE. 1.4 The measured X max distribution is reproduced weighting the distributions of different primary particles.The method assumes that the observed events N data are a mixture of N m pure mass samples with unknown fractions p j. The expected number of showers with X max into i-th bin is a ij = MC events from primary j into the i-th bin; N j MC = total number of MC events from primary j. The probability to observe n i events into the i-th bin is given by the product of Poisson distributions of mean ν i Primary fractions are determined maximizing the logarithm of P(n i ) with respect to p j Mass Composition from a logarithmic likelihood fit to X max distribution (LLF) Performances For different proton-iron mixing, N events have been randomly selected from proton and iron Monte Carlo data and the resulting samples have been analyzed. The whole procedure have been repeated many times. The input abundances are well reproduced by the methods in all cases, with a root mean square of the distribution of the reconstructed input fractions of less than 5%. Data reported by the Auger Collaboration at the ICRC 2007 have been analyzed in terms of proton and iron primaries and the measured Elongation Rate curve has been compared with that estimated considering, in each energy bin, the mean X max corresponding to the reconstructed mixture of LLF and MTA. Auger results have been confirmed with independent Monte Carlo techniques which can be corrected for the bias introduced by the analysis cuts applied and exploit a larger statistics avoiding very strong cuts. A set of observables define a parameter space populated with simulated cascades produced by different primaries. In each cell (h1, …, hn), the fraction of the population of primary i define the probability for a real shower falling into the cell to be initiated by a nucleus of species i. Multiparametric Analysis for the primary composition Reconstructed primary fractions are corrected for the mixing probabilities P i→j that an event of mass i is identified as primary j, and for the trigger-reconstruction-selection efficiency for each primary mass The primary fractions for a data set of N data showers is then given by the mean classification probability over the sample Mass composition is derived from the best choice of primary fractions that reproduce observed mean and variance of X max distribution using their expectation values (method of moments, MM). Modelling a data set of cosmic rays as a mixture of three primary masses (a, b and c) with relative abundances P a, P b and Pc = 1 - P a – P b, the expected mean shower maximum is where is the mean X max for simulated data set of the i-th species.The same applies to estimate the expected variance (ΔX exp ) 2. Assuming that the data set is so large that and Δ X max are statistically independent, in each energy bin, data could be fitted obtaining P a and P b. Composition analysis with the moments of X max distribution (MM)