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?)

To clarify, $\lambda=\operatorname{E}(X)=\operatorname{Var}(X)$.  The normal approximation is $N(\mu=\lambda,\sigma^2=\lambda)$ when $\lambda$ is "large".
$\lambda$ is often interpreted as an event arrival rate.
If you construct pseudorandom arrival times for the rate $\lambda_1 = \frac{1}{\text{1 hour}}$; you could "reuse" the constructed arrival times for a slower arrival process $\lambda_{24} = \frac{1}{\text{24 hour}}$ by scaling the pseudorandom intervals.
Defining $T_0$ as the interval start time, your arrival times for the faster process would stretch:  $$t_{i,24} = 24 ~(t_{i,1}-T_0)\quad .$$
In general,
$$t_{i,\lambda_2} = (t_{i,\lambda_1} - T_0)~ \frac{\lambda_1}{\lambda_2}\quad.$$
