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Additional resources for Attacks on Watermarking Systems

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Usually, one has to balance accuracy against efficiency. The smallest value recommended is k=10. (In fact, one can observe that k=12 has computational advantages). An alternative approach to generating normal variates (known as the direct approach) is the following. Let r1 and r2 be two uniformly distributed independent random numbers. Then 1 2 x1 = (-2 loge r1) cos 2πr2 1 2 x2 = (-2 loge r1) sin 2πr2 are two random variates from the standard normal distribution. This method produces exact results and the speed of calculations compares well with the Central Limit approach subject to the efficiency of the special function subroutines.

However, in most of the cases one will observe that the time a machine is operational varies. Also, the repair time may vary from machine to machine. If we are able to observe the operational times of a machine over a reasonably long period, we will find that they are typically characterized by a theoretical or an empirical probability distribution. Similarly, the repair times can be also characterized by a theoretical or empirical distribution. Therefore, in order to make the simulation model more realistic, one should be able to randomly numbers that follow a given theoretical or empirical distribution.

Rk. Since each ri is a uniformly distributed random number over the interval [0 ,1], we have that a+b 1 E(ri) = 2 = 2 (b-a)2 1 Var(ri) = 12 = 12 . Using the Central Limit theorem, we have that the sum Σri of these k random numbers approaches the normal distribution. That is ⎛k k ⎞ ⎟ , Σri ~ N⎜⎝2 , 12⎠ or Σri - k/2 k/ 12 ~ N(0, 1). 1) Now, let us consider the normal distribution with parameters µ and σ from which we want to generate normal variates. Let x be such a normal variate. Then x-µ σ ~ N(0,1).