Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.16 EXERCISES

tivariate Gaussian. For example, let us consider the quadratic form (multiplied

by 2) appearing in the exponent of (30.148) and write it asχ^2 n,i.e.

χ^2 n=(x−μ)TV−^1 (x−μ). (30.150)

From (30.149), we see that we may also write it as

χ^2 n=

∑n

i=1

y′i

2

λi

,

which is the sum ofnindependent Gaussian variables with mean zero and unit

variance. Thus, as our notation implies, the quantityχ^2 nis distributed as a chi-

squared variable of ordern. As illustrated in exercise 30.40, if the variablesXiare

required to satisfymlinear constraints of the form

∑n

i=1ciXi=0thenχ

2
ndefined
in (30.150) is distributed as a chi-squared variable of ordern−m.

30.16 Exercises

30.1 By shading or numbering Venn diagrams, determine which of the following are
valid relationships between events. For those that are, prove the relationship
using de Morgan’s laws.

(a) (X ̄∪Y)=X∩Y ̄.

(b)X ̄∪Y ̄=(X∪Y).

(c) (X∪Y)∩Z=(X∪Z)∩Y.

(d)X∪(Y∩Z)=(X∪Y ̄)∩Z ̄.

(e) X∪(Y∩Z)=(X∪Y ̄)∪Z ̄.

30.2 Given that eventsX, YandZsatisfy

(X∩Y)∪(Z∩X)∪(X ̄∪Y ̄)=(Z∪Y ̄)∪{[(Z ̄∪X ̄)∪(X ̄∩Z)]∩Y},
prove thatX⊃Y, and that eitherX∩Z=∅orY⊃Z.
30.3 AandBeach have two unbiased four-faced dice, the four faces being numbered
1, 2, 3, 4. Without looking,Btries to guess the sumxof the numbers on the
bottom faces ofA’s two dice after they have been thrown onto a table. If the
guess is correctBreceivesx^2 euros, but if not he losesxeuros.
DetermineB’s expected gain per throw ofA’s dice when he adopts each of the
following strategies:
(a) he selectsxat random in the range 2≤x≤8;
(b) he throws his own two dice and guessesxto be whatever they indicate;
(c) he takes your advice and always chooses the same value forx. Which number
would you advise?

30.4 Use the method of induction to prove equation (30.16), the probability addition
law for the union ofngeneral events.
30.5 Two duellists,AandB, take alternate shots at each other, and the duel is over
when a shot (fatal or otherwise!) hits its target. Each shot fired byAhas a
probabilityαof hittingB, and each shot fired byBhas a probabilityβof hitting
A. Calculate the probabilitiesP 1 andP 2 , defined as follows, thatAwill win such
a duel:P 1 ,Afires the first shot;P 2 ,Bfires the first shot.
If they agree to fire simultaneously, rather than alternately, what is the proba-
bilityP 3 thatAwill win, i.e. hitBwithout being hit himself?