TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW07.5

Law of large numbers

Visualize the law of the large numbers assuming exponential distributed random variables. Let $X_{1},X_{2},\dots$ be i.i.d. random variables, with $X_{1}\sim {\text{exp}}(\lambda )$.

For $m=1,2,\dots$ the mean of the first $m$ random variables is given as

$${\bar {X}}_{m}={\frac {1}{m}}\sum _{i=1}^{m}X_{i},\quad m\in \mathbb {N} .$$

Further, for $n\in \mathbb {N}$ a sequence of means is given as ${\bar {X}}_{1},{\bar {X}}_{2},\dots ,{\bar {X}}_{n}$. Assume $\lambda =0.5$ and $n=200$ and plot 20 realized sequences of means in one graph. Mark the expectation of $X_{1}$. Comment on the obtained result.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

# Visualize the law of the large numbers assuming exponential distributed random variables. 
# Let X1, X2, . . . be i.i.d. random variables, with X1 ∼ exp(λ).

n <- 200
lambda <- 0.5
exp <- 1 / lambda
seqs <- 20
color_palette <- rainbow(seqs)


plot(exp, type = "n", xlab = "n", ylab = expression(bar(X)[n]), xlim=c(1, 200), ylim=c(0, 2*exp))
abline(h = exp, lty=2)

for (i in 0:seqs) {
  sequence <- c()
  
  for (m in seq(1:n)) {
    X <- rexp(m, lambda)
    sequence[m] = (1 / m) * sum(X)
  }
  
  lines(1:n, sequence, col=color_palette[i])
}