90 Minutes

Angry Birds Holiday Special - download pdf or read online

By Kari Korhonen, Corrado Mastantuono

While Terence, the most important of the indignant Birds, is installed cost of the North Pole all his desires come precise. yet, being in control of providing programs to each sturdy little lady and boy on the planet seems to be a more challenging task than he may have imagined and his goals speedy flip to nightmares! Will the remainder of the indignant Birds manage to are available in time to avoid wasting the day? And the place is Santa Claus besides? discover during this over-sized vacation certain!

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Some Background and Preliminaries 38 for all N > k + 1 and k < n < N - 1. Hence, by Bayes's formula, N-1 N) _ pp pp ((Nk} = n)1Pp ((N1}n < N - n). n-k - n) < pfq for 1 < n < N, Therefore, since Pp(((') < N (c{k+1} < N) lim P < N p Ilpp( k} = n) lim p V q N-4oo N-*oo n=k lim Pp ((Nk} < N), p p V q N-*oo and so lim Pp ((Nk+1 ) < N) < N-3oo lim jp (c{k} < N) p - p V g N--}oo . n < N - n), n.. =k and therefore m Pp (({k+1 } N M-1 P Epp( {kj N) > p V q - n) n=k pVgpp((Mk}1

Hence, R2r n B22111 54- 0 if and only if r < All - m and R- n BZ [ -- R. n C2m (2r). 1-m) = 0) = 4_11,1 2m m 2(111 - m) M-m 1. Some Background and Preliminaries 26 for all M > 1 and 0 < m < M. A/1 > 1 and 0 < m < M. 27) satisfy (*) and (**). There is nothing to do when m E {0, 2M}. To prove it when I < m < M, observe that {W2m=0&W2M0}= {W2m= 0}(1 {W2(M-m)-W2m=0} and therefore that uM = P(W2m = 0 & W2M = 0). 26), m m um1 = >P(R2r)P(W2(mr) = 0 & W2(A1-r) = 0) = E P(R2r)um r r-l . r=1 Similarly, because {W2(ll'I-m) = 0 & W2M = 0) _ {W2(1ti1--m) = 0} n {W2M - W2(1U1-ni) - 01, EM 1"'" P(R2r)u l -r, and so {uml : I < m < M} satisfies (**).

Knowing that, with probability close to 1, the size of }WA; is no larger than a large constant times N21, it is reasonable to look more closely and to ask about the probability that WN - N-- 2 WN lies in an interval. The 1. 18 (De Moivre). 19) 2 e2 2 (2-7r) a where the convergence is uniform with respect to a, b E [-oo, oo] with a < b. (0 < W2N < x) -i (27r) (*) 2 e N-400 d, fo where the convergence is uniform with respect to x in bounded subsets of (0, o©). )2 - k)! 7) IIj=1 (I+ NO x) = ---x -- E(x) for Jxj < 1, and observe that X xn E(x)l n=2 < n 2 In particular, jE(x) x2 for [xj and I < k < (2N)2R, then 2.

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Angry Birds Holiday Special by Kari Korhonen, Corrado Mastantuono


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