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Mathematical Methods for Physics and Engineering - Matematica.NET

Mathematical Methods for Physics and Engineering - Matematica.NET

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30.16 EXERCISES30.11 A boy is selected at r<strong>and</strong>om from amongst the children belonging to families withn children. It is known that he has at least two sisters. Show that the probabilitythat he has k − 1brothersis(n − 1)!(2 n−1 − n)(k − 1)!(n − k)! ,<strong>for</strong> 1 ≤ k ≤ n − 2 <strong>and</strong> zero <strong>for</strong> other values of k. Assume that boys <strong>and</strong> girls areequally likely.30.12 Villages A, B, C <strong>and</strong> D are connected by overhead telephone lines joining AB,AC, BC, BD <strong>and</strong> CD. As a result of severe gales, there is a probability p (thesame <strong>for</strong> each link) that any particular link is broken.(a) Show that the probability that a call can be made from A to B is1 − p 2 − 2p 3 +3p 4 − p 5 .(b) Show that the probability that a call can be made from D to A is1 − 2p 2 − 2p 3 +5p 4 − 2p 5 .30.13 A set of 2N + 1 rods consists of one of each integer length 1, 2,...,2N,2N +1.Three, of lengths a, b <strong>and</strong> c, are selected, of which a is the longest. By consideringthe possible values of b <strong>and</strong> c, determine the number of ways in which a nondegeneratetriangle (i.e. one of non-zero area) can be <strong>for</strong>med (i) if a is even,<strong>and</strong> (ii) if a is odd. Combine these results appropriately to determine the totalnumber of non-degenerate triangles that can be <strong>for</strong>med with the 2N + 1 rods,<strong>and</strong> hence show that the probability that such a triangle can be <strong>for</strong>med from ar<strong>and</strong>om selection (without replacement) of three rods is(N − 1)(4N +1).2(4N 2 − 1)30.14 A certain marksman never misses his target, which consists of a disc of unit radiuswith centre O. The probability that any given shot will hit the target within adistance t of O is t 2 ,<strong>for</strong>0≤ t ≤ 1. The marksman fires n independendent shotsat the target, <strong>and</strong> the r<strong>and</strong>om variable Y is the radius of the smallest circle withcentre O that encloses all the shots. Determine the PDF <strong>for</strong> Y <strong>and</strong> hence findthe expected area of the circle.The shot that is furthest from O is now rejected <strong>and</strong> the corresponding circledetermined <strong>for</strong> the remaining n − 1 shots. Show that its expected area isn − 1n +1 π.30.15 The duration (in minutes) of a telephone call made from a public call-box is ar<strong>and</strong>om variable T . The probability density function of T is⎧⎪⎨ 0 t

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