Let T be the time until a radioactive particle decays, and suppose (as is often done in

Chapter 5, Problem 40

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Let T be the time until a radioactive particle decays, and suppose (as is often done in physics and chemistry) that T Expo(). (a) The half-life of the particle is the time at which there is a 50% chance that the particle has decayed (in statistical terminology, this is the median of the distribution of T). Find the half-life of the particle. (b) Show that for a small, positive constant, the probability that the particle decays in the time interval [t, t + ], given that it has survived until time t, does not depend on t and is approximately proportional to . Hint: ex 1 + x if x 0. (c) Now consider n radioactive particles, with i.i.d. times until decay T1,...,Tn Expo(). Let L be the first time at which one of the particles decays. Find the CDF of L. Also, find E(L) and Var(L). (d) Continuing (c), find the mean and variance of M = max(T1,...,Tn), the last time at which one of the particles decays, without using calculus. Hint: Draw a timeline, apply (c), and remember the memoryless property.