Exercise 1 (Deterministic colouring) Let \(\Pi\) a Poisson point process on \([0, \infty)\). Suppose we colour the first arrival blue, and then next arrival red, and continue colouring the points in this alternating fashion. Consider the point processes \(\Gamma\) formed by considering only the blue points. Is this a Poisson point process? Explain.
Solution. No. The interarrival times between two blue points is given by the sum of two exponential random variables, and is no longer an exponential.
Exercise 2 (Shop keeper) Suppose we model the number of customers that arrive at a high street shop on at particular day by a Poisson process of intensity \(\lambda >0\), where \(\lambda\) is measured in customers per hour. We wish to estimate \(\lambda\). Suppose the shop is really high-end and on some days has no customers, on its \(6\) hours of operations. The shop keeper only keeps track of whether she had has any customers are not; that is, her records \(x = (x_1, \ldots, x_n)\) are a binary sequence. Find a consisent estimtor for \(\lambda\)
Solution. Let \(X_i\) be the binary Bernoulli random variables that corresponds to whether a customer arrived on the \(i\)th day. Note that \(\mathbb{P}(X_i = 0) = e^{-6\lambda}\) and \(\mathbb{P}(X_i = 1) = 1 - e^{-6\lambda}\). Assume that these are independent random variables, the law of large numbers gives that
\[ n^{-1}(X_1 + \cdots X_n) \to \mathbb{E}X_1 = 1 - e^{-6\lambda},\]
as \(n \to \infty\).
Thus setting \[T_n := -\frac{1}{6}\log( 1- n^{-1}(X_1 + \cdots+ X_n)),\]
we also know that \(T_n \to \lambda\), as \(n \to \infty\), since \(\log\) is continuous.
Exercise 3 Suppose we \(\Pi\) is a Poisson point process on \([0, \infty)\) of intensity \(\lambda\). Using the construction of \(\Pi\) as exponential inter-arrival times, prove that conditioned on the event that the unit interval contains exactly one point, the distribution of the its location is uniform.
Solution. Let \(X_1\) and \(X_2\) be independent exponential random variables of rate \(\lambda\) representing the inter-arrival times. Let \(U\) be the location of the point in question. We are asked to compute
\[\mathbb{P}(U \leq x | X_1 < 1, X_2+X_1 >1)=\mathbb{P}(X_1 \leq x | X_1 < 1, X_2+X_1 >1)\]
for \(0 < x <1\).
Since \(X_1\) and \(X_2\) are independent, we have
\[\begin{eqnarray*}
\mathbb{P}(X_1 < x, X_2+X_1 >1) &=& \int_0 ^x \lambda e^{- \lambda x_1} \Big( \int_{1-x_1} ^{\infty} \lambda e^{ -\lambda x_2} dx_2 \Big)dx_1 \\
&=&\int_0 ^x \lambda e^{-\lambda x_1} e^{-\lambda(1-x_1)}dx_1 \\
&=& x \lambda e^{-\lambda}.
\end{eqnarray*}\]
Hence, using this formula in the numerator, and also in the denominator with \(x=1\), we have
\[\mathbb{P}(U \leq x | X_1 < 1, X_2+X_1 >1) = \frac{x \lambda e^{-\lambda}}{ \lambda e^{-\lambda}} =x\]
as desired.
Exercise 4 (Random deletion) Simulate a Poisson process of intensity \(\lambda=2\), say with \(10000\) arrivals. Delete each arrival independently with probability \(p=\tfrac{1}{2}\) to from a new thinned process. Plot a histogram of the inter-arrival times of the thinned process. What should you see? Why?
Solution. The histogram approximates the exponential density with rate \(1\), since we know that the resulting thinned process is still a Poisson process with intensity \(p \lambda=1\).
inter=rexp(10000, 2)
arr = cumsum(inter)
coin = rbinom(length(arr), 1, 0.5)
for (i in 1:length(arr)) if (coin[i]==0){arr[i]<- 0}
arr<-setdiff(arr, 0)
interthin = arr[1]
for (i in 1: length(arr)-1){interthin <- c(interthin, arr[i+1] - arr[i]) }
x = seq(0, max(interthin)+1, by=0.1)
hist(interthin, prob=T, x)
curve(dexp(x), add=T)
![](data:image/png;base64,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)
Exercise 5 (Poisson on a disc) Simulate a Poisson point process of intensity \(100\) on a disc.
Solution. We simulate a single point on the disc, by simulating a uniform point on a square, and keep it if lands inside an inscribed disc, or repeat otherwise. Then we implent the characterization of Poisson point processes as a Poisson number of uniform points. Note a disc has area \(\pi r^2\).
point <- function(){
z=2
while(length(z)==1){
x = 2*runif(1) -1
y = 2*runif(1) -1
if (x^2 + y^2 <1) { z<-c(x,y)}
}
z
}
re=replicate(rpois(1,100*pi), point())
plot(re[1,], re[2,], xlim=c(-1.1,1.1), ylim=c(-1.1,1.1), asp=1)
![](data:image/png;base64,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)