Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.16 EXERCISES

30.18 A particle is confined to the one-dimensional space 0≤x≤a, and classically
itcanbeinanysmallintervaldxwith equal probability. However, quantum
mechanics gives the result that the probability distribution is proportional to
sin^2 (nπx/a), wherenis an integer. Find the variance in the particle’s position
in both the classical and quantum-mechanical pictures, and show that, although
they differ, the latter tends to the former in the limit of largen, in agreement
with the correspondence principle of physics.
30.19 A continuous random variableXhas a probability density functionf(x); the
corresponding cumulative probability function isF(x). Show that the random
variableY=F(X) is uniformly distributed between 0 and 1.
30.20 For a non-negative integer random variableX, in addition to the probability
generating function ΦX(t) defined in equation (30.71), it is possible to define the
probability generating function

ΨX(t)=

∑∞

n=0

gntn,

wheregnis the probability thatX>n.

(a) Prove that ΦXand ΨXare related by

ΨX(t)=

1 −ΦX(t)

1 −t

.

(b) Show thatE[X]isgivenbyΨX(1) and that the variance ofXcan be

expressed as 2Ψ′X(1) + ΨX(1)−[ΨX(1)]^2.

(c) For a particular random variableX, the probability thatX>nis equal to

αn+1,with0<α<1. Use the results in (b) to show thatV[X]=α(1−α)−^2.

30.21 This exercise is about interrelated binomial trials.

(a) In two sets of binomial trialsTandt, the probabilities that a trial has a

successful outcome arePandp, respectively, with corresponding probabilites

of failure ofQ=1−Pandq=1−p. One ‘game’ consists of a trialT,

followed, ifTis successful, by a trialtand then a further trialT.Thetwo

trials continue to alternate until one of theT-trials fails, at which point the

game ends. The scoreSfor the game is the total number of successes in the

t-trials. Find the PGF forSand use it to show that

E[S]=

Pp

Q

,V[S]=

Pp(1−Pq)

Q^2

.

(b) Two normal unbiased six-faced diceAandBare rolled alternately starting

withA;ifAshows a 6 the experiment ends. IfBshows an odd number no

points are scored, if it shows a 2 or a 4 then one point is scored, whilst if

it records a 6 then two points are awarded. Find the average and standard

deviation of the score for the experiment and show that the latter is the

greater.

30.22 Use the formula obtained in subsection 30.8.2 for the moment generating function
of the geometric distribution to determine the CGF,Kn(t), for the number of
trials needed to recordnsuccesses. Evaluate the first four cumulants, and use
them to confirm the stated results for the mean and variance, and to show that
the distribution has skewness and kurtosis given, respectively, by

2 −p

√

n(1−p)

and 3 +

6 − 6 p+p^2

n(1−p)

.