Further Properties of Poisson Processes

Compound Poisson Process

n_rep = 10000

t = 60

lambda = 12

mu = 5

sigma = 1.5

N = rep(NA, n_rep)
X = rep(NA, n_rep)

for (i in 1:n_rep) {
  N[i] = rpois(1, lambda * t)
  
  X[i] = sum(rnorm(N[i], mu, sigma))
}

hist(X)

mean(X)

[1] 3599.934

sd(X)

[1] 139.1417