Simulation Procedures

 Suppose that we want to simulate a homogeneous Poisson process with intensity $\lambda$ in the bounded region B where $0 < \nu(B) < \infty$. In achieving this goal we can use the conditional uniformity property (2.1). Thus, we condition on the total number of points X(B) in the set B. The simulation therefore consists of two steps. First, the Poisson distributed random variable X(B) is simulated. If this is done, say with the result X(B)=n, then the second step follows easily by using (2.1), if one knows how to simulate n independent and uniformly distributed points in the region B. Note that it is straightforward to simulate n independent random points uniformly distributed in [0,1]²: if $Z_1,Z_2,\ldots, Z_{2n-1},Z_{2n}$ is a sequence of independent random numbers uniformly distributed in [0,1], then the n random points $X_1=(Z_1,Z_2),\ldots,X_n=(Z_{2n-1},Z_{2n})$ are independent and uniformly distributed in [0,1]². Next, translation and scaling can be used to generate a sequence of n independent random points uniformly distributed in an arbitrary fixed rectangle. Now, in order to simulate n independent random points uniformly distributed in a more complex bounded region B we can apply different methods. One is to consider a rectangle $B^{\prime}$which contains B and to generate so many points in $B^{\prime}$until n of them fall in B. These n random points are then independent and uniformly distributed in B. In Figure 5 this method is used to simulate n=28 random points uniformly distributed in the region B.

 

Figure 5:  Simulation

Another way is to approximate B by a finite number of disjoint rectangles $B_1,\ldots, B_k$ and to simulate n independent and uniformly distributed points in the union $\bigcup_{i=1}^k B_i$. This is realized by choosing one of these k rectangles with probability proportional to its area, simulating a random point uniformly distributed in this rectangle, and repeating this procedure n times. If the boundary of B is rather rough, it can be useful to combine the above described methods.