Mathematical Methods for Physics and Engineering - Matematica.NET

30.16 EXERCISEStivariate Gaussian. For example, let us consider the quadratic form (multipliedby 2) appearing in the exponent of (30.148) and write it as χ 2 n,i.e.χ 2 n =(x − µ) T V −1 (x − µ). (30.150)From (30.149), we see that we may also write it asn∑χ 2 y ′ 2in = ,λ ii=1which is the sum of n independent Gaussian variables with mean zero and unitvariance. Thus, as our notation implies, the quantity χ 2 n is distributed as a chisquaredvariable of order n. As illustrated in exercise 30.40, if the variables X i arerequired to satisfy m linear constraints of the form ∑ ni=1 c iX i =0thenχ 2 n definedin (30.150) is distributed as a chi-squared variable of order n − m.30.16 Exercises30.1 By shading or numbering Venn diagrams, determine which of the following arevalid relationships between events. For those that are, prove the relationshipusing de Morgan’s laws.(a) ( ¯X ∪ Y )=X ∩ Ȳ .(b) ¯X ∪ Ȳ = (X ∪ Y ).(c) (X ∪ Y ) ∩ Z =(X ∪ Z) ∩ Y .(d) X ∪ (Y ∩ Z) =(X ∪ Ȳ ) ∩ ¯Z.(e) X ∪ (Y ∩ Z) =(X ∪ Ȳ ) ∪ ¯Z.30.2 Given that events X,Y and Z satisfy(X ∩ Y ) ∪ (Z ∩ X) ∪ ( ¯X ∪ Ȳ )=(Z ∪ Ȳ ) ∪{[( ¯Z ∪ ¯X) ∪ ( ¯X ∩ Z)] ∩ Y },prove that X ⊃ Y , and that either X ∩ Z = ∅ or Y ⊃ Z.30.3 A and B each have two unbiased four-faced dice, the four faces being numbered1, 2, 3, 4. Without looking, B tries to guess the sum x of the numbers on thebottom faces of A’s two dice after they have been thrown onto a table. If theguess is correct B receives x 2 euros, but if not he loses x euros.Determine B’s expected gain per throw of A’s dice when he adopts each of thefollowing strategies:(a) he selects x at random in the range 2 ≤ x ≤ 8;(b) he throws his own two dice and guesses x to be whatever they indicate;(c) he takes your advice and always chooses the same value for x. Which numberwould you advise?30.4 Use the method of induction to prove equation (30.16), the probability additionlaw for the union of n general events.30.5 Two duellists, A and B, take alternate shots at each other, and the duel is overwhen a shot (fatal or otherwise!) hits its target. Each shot fired by A has aprobability α of hitting B, and each shot fired by B has a probability β of hittingA. Calculate the probabilities P 1 and P 2 , defined as follows, that A will win sucha duel: P 1 , A fires the first shot; P 2 , B fires the first shot.If they agree to fire simultaneously, rather than alternately, what is the probabilityP 3 that A will win, i.e. hit B without being hit himself?1211