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5. Alpha particles are emitted from a radioactive material according to a Poisson process with intensity A. Suppose that a Geiger counter registers an emitted particle with probability 3), independently for each particle. Let the value of the process X (t) be the number of counted particles in the interval (0, t]. (a) Show that X(t) is a Poisson process with intensity pA. (13p) (b) Let N1 denote the number of emitted particles until the rst particle is counted, and let T1 be the time until the rst particle is counted. Derive the distributions of N1 and T1. (7p)