Transform "Standard Poisson" to any Poisson If I use a RNG to generate a set of standard normal draws $(Z_i)$, anyone can get samples from a normal distribution with $(\mu, \sigma^2)$ of their choosing via $X_i=Z_i.\sigma+\mu$. This doesn't require them to do any random number generating themselves (i.e. they don't need a RNG)
Is the same thing possible for Poisson distributions? i.e. If I generate a set of Poisson draws with $\lambda=1$ (or some other $\lambda$), is it possible for another person to get Poisson samples with whatever $\lambda$ they want, without having to use an RNG?
(I realise you could use the normal approximation $N(\lambda,\sqrt\lambda)$ if $\lambda$ is large, but what if this isn't the case?)

If you have a lot of independent Poisson values when $\lambda=1$ then is possible to construct independent Poisson values when $\lambda^\prime=n$ for any positive integer $n$ by adding $n$ of your original values together
To take Stephan Kolassa's example with $n=5$, by simulating $500000$ cases of $\lambda=1$ to generate $100000$ cases of $\lambda^\prime=5$, and then comparing to the actual distribution in red, you could use this R code:
set.seed(2021)
n <- 5 
samplesize <- 10^5
x1 <- rpois(n * samplesize, lambda=1)
xn <- rowSums(matrix(x1, ncol=n))
table(xn)
# xn
#    0     1     2     3     4     5     6     7     8     9    10    11    12 
#  693  3403  8359 14001 17540 17385 14758 10594  6452  3642  1724   869   367 
#   13    14    15    16    17    18 
#  134    54    14     9     1     1 
plot(table(xn) / samplesize)
m <- min(xn):max(xn)
points(m, dpois(m, lambda=n), col="red")


which is pretty close for a simulation
