By George Whitefield

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**Extra resources for A letter to ... dr. Durell ... occasioned by a late expulsion of six students from Edmund hall**

**Sample text**

For each i, determine whether si ≤ P (˜ is the case, then keep r˜i ; otherwise delete it from the sequence of random numbers. The remaining r˜i are then distributed according to the distribution function P . The von Neumann method can be used for creating any desired probability distribution. However, it often consumes a lot of computing time for generating random numbers that are thrown away later on, especially if the desired distribution peaks in some areas of the interval and nearly vanishes in other areas.

Above all, detailed mathematical insight into the proposed optimization problem is needed to determine appropriate branching properties and good lower-bound functions. Sometimes it is better to work with a “wide tree”, i. , a tree with many branches going out from each node; for other cases, it is advantageous to work with a “deep tree”, i. , a tree that splits into only a few branches at each node such that more branching layers are needed. A special case of a deep tree is the binary tree in which each branch splits into two subbranches.

A natural choice is N = 12, as then the standard deviation of the resulting Gaussian distribution is 1: the expectation value of a uniformly distributed random number ri in the interval [0; 1] is ri = 12 , ri2 = 13 , ri rj = 14 for i = j. Therefore, σ 2 (gi ) = −6 + = 12 j=1 ri,j 12 j=1 ri,j 2 2 − −6 + 12 j=1 ri,j − 12 2 = (ri,1 + ri,2 + . . + ri,12 ) 12 j=1 ri,j 2 + 36 − 0 − 12 × 6 + 36 = 12 ri,j 2 + 2 × (11 + 10 + . . + 1) × ri,j × ri,k=j − 36 = 12 × 1 3 +2× 11×12 2 = 4 + 33 − 36 = 1 . 12) 40 3 Monte Carlo Note that the disadvantage of summing up only 12 uniformly distributed random numbers is that only Gaussian random numbers in the interval [−6; 6] can be obtained.