Mathematical Methods for Physics and Engineering - Matematica.NET

30.16 EXERCISES30.18 A particle is confined to the one-dimensional space 0 ≤ x ≤ a, and classicallyitcanbeinanysmallintervaldx with equal probability. However, quantummechanics gives the result that the probability distribution is proportional tosin 2 (nπx/a), where n is an integer. Find the variance in the particle’s positionin both the classical and quantum-mechanical pictures, and show that, althoughthey differ, the latter tends to the former in the limit of large n, in agreementwith the correspondence principle of physics.30.19 A continuous random variable X has a probability density function f(x); thecorresponding cumulative probability function is F(x). Show that the randomvariable Y = F(X) is uniformly distributed between 0 and 1.30.20 For a non-negative integer random variable X, in addition to the probabilitygenerating function Φ X (t) defined in equation (30.71), it is possible to define theprobability generating function∞∑Ψ X (t) = g n t n ,where g n is the probability that X>n.(a) Prove that Φ X and Ψ X are related byΨ X (t) = 1 − Φ X(t).1 − t(b) Show that E[X] is given by Ψ X (1) and that the variance of X can beexpressed as 2Ψ ′ X (1) + Ψ X(1) − [Ψ X (1)] 2 .(c) For a particular random variable X, the probability that X>nis equal toα n+1 ,with0