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false positives /false negatives Mistakes randomly made (presumably with low probability) by some randomized algorithms that work in a fixed amount of time (p 377) Fermat's Little Theorem States that if P is prime and 0 < A < P, then A P I = I (mod P ) It is necessary but not sufficient to establish primality (p 379) full-period linear congruential generator A random number generator that has period M - l (p 369) linear congruential generator A good algorithm for generating uniform distributions (p 368) negative exponential distribution A form of distribution used to model the time between occurrences of random events Its mean equals its variance (p 373) period The length of the sequence until a number is repeated A random number generator with period P generates the same random sequence of random numbers after P iterations (p 369) permutation A permutation of 1 , 2, , N is a sequence of N integers that includes each of I , 2, , N exactly once (p 365) Poisson distribution A distribution that models the number of occurrences of a rare event (p 37 1 ) pseudorandom numbers Numbers that have many properties of random numbers Good generators of pseudorandom numbers are hard to find (p 366) random permutation A random arrangement of N items Can be generated in linear time using one random number per item (p 374) randomized algorithm An algorithm that uses random numbers rather than deterministic decisions to control branching (p 376) seed The initial value of a random number generator (p 369) trial division The simplest algorithm for primality testing It is fast for small (32-bit) numbers but cannot be used for larger numbers (p 378) uniform distribution A distribution in which all numbers in the specified range are equally likely to occur (p 367) witness to compositeness A value of A that proves that a number is not prime, using Fermat's Little Theorem (p 379).



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1 Using an initial seed of zero gives bad random numbers 2 Inexperienced users occasionally reinitialize the seed prior to generating a random permutation This action guarantees that the same permutation will be repeatedly produced, which is probably not intended 3 Many random number generators are notoriously bad; for serious applications in which long sequences of random numbers are required, the linear congruential generator is also unsatisfactory 4 The low-order bits of linear congruential generators are known to be somewhat nonrandom, so avoid using them For example, randomInt ( ) %2 is often a bad way to flip a coin 5 When random numbers are being generated in some interval, a common error is to be slightly off at the boundaries and either allow some number outside the interval to be generated or not allow the smallest number to be generated with fair probability 6 Many random permutation generators do not generate all permutations with equal likelihood As discussed in the text, our algorithm is limited by the random number generator 7 Tinkering with a random number generator is likely to weaken its statistical properties

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Most of the code in this chapter is available Contains the class interface for Random, as shown in Figure 101 Randomcpp Contains the Random class implementation Permutecpp Contains the permute routine shown in Figure 106 Contains the primality-testing routine shown in FigMathcpp ure 108 and the math routines presented in Section 84

Filling Out the Form To best understand how secondary data sources work when you re lling out a form, let s continue with the XML le data connection we created in the preceding chapter Figure 76 shows how the form looks after querying the XML data

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In Short 101 For the random-number generator described in the text, determine the first 10 values of state,assuming that it is initialized with a value of 1

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