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plPoisson

Definition 14   Given $\lambda\in I\!\!R^\ast$ a plPoisson on $\Omega=\{X_1\}$ is a kernel where the compute function is defined as follows:

$$Compute(\omega) = Poisson(\omega,\lambda)$$ (3.10)

$$Poisson(\omega,\lambda) = \frac{\exp(-\lambda)\lambda^x}{x!}$$ (3.11)

Example 7   Here we define a plPoisson kernel on $\Omega =\{[0,50]\}$ given $\lambda =5$.

図 3.7: A poisson kernel on $\Omega =\{[0,50]\}$ and $\lambda =5$.

The plPoisson kernel is constructed by

  plPoisson Px(X,5);

The kernel graph is given by Figure 3.7 and the program output of shows as follow:

X = {x} with x in [0,1,...,50]
P(x) = plPoisson(x,5)

Generating 5 random values
draw # 0 = { x=8 }
draw # 1 = { x=9 }
draw # 2 = { x=2 }
draw # 3 = { x=4 }
draw # 4 = { x=5 }

Generating 5 best values
best # 0 = { x=5 }
best # 1 = { x=4 }
best # 2 = { x=4 }
best # 3 = { x=5 }
best # 4 = { x=4 }

Examples of compute
compute({ x=0 } )= 0.00673795
compute({ x=10 } )= 0.0181328
compute({ x=25 } )= 1.29459e-10
compute({ x=30 } )= 2.36574e-14

Notice that, the best function returns two different values. In effect, if $\mu \in Z^\ast$ then $Poisson(\omega,\lambda)$ has with two minimum at $\lambda ~^+_- 1$