Question

In 17]:import numpy as np import matplotlib.pyplot as plt Implement three different methods to simulate a Poisson process (Nog 100 with parameter λ = 0.1 on the time interval [0, 100]. For each method, plot a trajectory of your simulated process 1. Method 1:

Answer 1

SOLUTION ;-

-;GIVEN DATA;-

implement three diffrent methods to simlate a poisson process (Nt) with parameter =0.1 on the interval [0,100]

import numpy as np

import matplotlib.pyplot as plt

method 1 -

import numpy as np import matplotlib.pyplot as plt

N<-10
A<-vector(mode="integer",length = N)
A[1]=0

# parameter lambda

X<-rexp(n =N,rate = 2)
for(n in 1:N)
A[n]=sum(X[1:n])
A<-cumsum(X)
n_func <- function(t, A) sapply(t, function(t) sum(A < t))
t_series <- seq(0, max(S), by = max(A)/N)

#Plot of the trajectory

par(mfrow=c(1,2))
plot(t_series, n_func(t_series, A),
type = "a",
ylab=expression(N[t]),
xlab="t",las=1,
cex.lab=0.8,main="Poisson Process",cex.axis=0.8)
grid()
abline(v = A,col="red",lty=2)
plot(t_series, n_func(t_series, A),
type = "a",
ylab=expression(N[t]),
xlab="t",las=1,
cex.lab=0.8,
main="Poisson Process",cex.axis=0.8)
grid()
abline(v = t_series,col="blue",lty=4)

method 2-

  

import numpy as np import matplotlib.pyplot as plt

# Prepare data
N = 100
lambdas = [1, 2, 5]
X_T = [np.random.poisson(lam, size=N) for lam in lambdas]
S = [[np.sum(X[0:i]) for i in xrange(N)] for X in X_T]
X = np.linspace(0, N, N)

# Plot trajectory
graphs = [plt.step(X, S[i], label="Lambda = %d"%lambdas[i])[0] for i in xrange(len(lambdas))]
plt.legend(handles=graphs, loc=2)
plt.title("Poisson Process", fontdict={'fontname': 'Times New Roman', 'fontsize': 21}, y=0.1)
plt.ylim(0)
plt.xlim(0)
plt.show()