The probability density function (PDF) describing the time (t) between the creation and decay of an unstable elementary particle is given by the exponential distribution: f(t) = ae-t/ for t20 with T, a constant Follow TAYLOR, Section 5.2, and use the following definite integrals. (Use no other information.) f(t) = 0 for t < 0 and 00 In = u" e-udu = n! (for integer n = 0,1, 2, ...) with u = Find each of the following quantities. (You must explicitly show how the given integral is used for each and every quantity. Even the probability can be found easily with a change of variables!) Normalization constant (a) in terms of the lifetime (7) Average time for a particle to decay (in terms of the lifetime) Variance (and from this the standard deviation) Probability for a particle to live greater than three lifetimes [Use lo.] [Use I1.] [Use l2, 1, 1o.] [Shift variable, then lo.]