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3) The simplest case is g(x) = x − y, hence x0 = y and |g (x0 )| = 1. 3) that by an additive change of the variable, x → x − y, the Dirac delta yields the functional value corresponding to a translation along the abscissa: ∞ −∞ ∞ f (x)δ(x − y) dx = f (y), −∞ f (x − y)δ(x) dx = f (−y). We imagine that the Dirac delta “combs” the real axis and thereby “samples” the function f at x = y. When g has several simple zeros {x0 , x1 , . . 3) must be considered in the vicinity of each zero separately: ∞ −∞ n f (x)δ (g(x)) dx = i=0 f (xi ) .

Sm ≈ i=1 P(Si |Dj )P(Dj ) m n i=1 k=1 P(Si |Dk )P(Dk ) . Inevitably, the assumption of symptom independence is quite coarse: given the presence of the disease, the probability of two symptoms appearing simultaneously is larger than the product of probabilities of individual symptoms. ) A particle moves along the real axis, starting at the origin (x = 0). Consecutive random collisions uniformly spaced in time send it one step to the left (−1) or to the right (+1) with probabilities 1/2 either way.

6184. ② Let P(k) denote the probability that precisely k blood types will be found in the chosen four. 0687. /(1! /(2! ) = 6. 5973. 5 Problems 21 The calculation for k = 3 is tedious and is best avoided by calculating the probability for k = 4, which is P(4) = 4! 3163. 3 Independence of Events in Particle Detection Two detectors are used to detect charged particles with different parities (mirror symmetries of their wave-functions): pions (π + and π − ) and kaons (K + and K − ), all possessing negative parity, as well as protons (p), deuterons (d) and 3 He and 4 He nuclei, all of which have positive parities.

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