Distribution of uniformly sampled points in a specific area with a Poisson distributed number of samples

Question

Assume I have a Poisson distribution that draws the number $n$ of points to sample: $\mathcal{P}_{\lambda}(x)$. Then, I sample $n$ points from a uniform distribution $\mathcal{U}_{[0,4]}(x)$

However, in the end, I am only interested in the points in [0, 4/d], and I do not want to sample all the unnecessary points. So, my idea was to sample $\tilde{n} = \mathcal{P}_{\lambda/d}(x)$ points from $\mathcal{U}_{[0,4/d]}(x)$ or sample $n' = \mathcal{P}_{\lambda}(x)/d$ points from $\mathcal{U}_{[0,4/d]}(x)$.

Are the results equal? Can I do so?

I created a short Matlab example:

clc
clear all
close all

lambda_7 = 7;
num_trials = 10000;
r74 = 0;
for i = 1:num_trials
    r7(i) = 0;
    for m = 1:poissrnd(lambda_7)
        if rand(1)*4 < 1
            r7(i) = r7(i) + 1;
        end
    end
    r74(i) = poissrnd(lambda_7/4.);
    r7b(i) = round(poissrnd(lambda_7)/4.);
end
avg7 = sum(r7) / num_trials
avg74 = sum(r74) / num_trials

figure
hold on
h7 = histogram(r7)
h74 = histogram(r74)

It points out that both could yield the same distribution.

(Sorry for the bad title, feel free to make it more concise.)

Edit: Matlab code revised.

Answer 1

Your second option of sampling $n' = \mathcal{P}_{\lambda}(x)/d$ points from $\mathcal{U}_{[0,4/d]}(x)$ should not work. $n'$ will have the wrong distribution in general (and sometimes may not be an integer) and will not have the same distribution as $\tilde n$.

So instead you could choose $n$ from $\mathcal{P}_{\lambda}(x)$ and then $n'$ by choosing it from the correct distribution, which is binomial with parameters $n$ and $\frac1d$, and then sample that many points from $\mathcal{U}_{[0,4/d]}(x)$. This will not be faster than your method for $\tilde n$ but might be easier for some people to understand.