Distribution of Poisson variable choose 2 I have a random Poisson variable, $$X\sim Po(\lambda)$$
Also, the value of $X$ determines the value of $Y$:
$$X=k\Rightarrow Y={k\choose{2}}= \frac{k^{2}-k}{2}$$
I guess there is no elegant way of writing a PMF, other than:
$$\text{Pr}\left[Y={k\choose{2}}\right]=\text{Pr}\left[X=k\right]$$ 
What more can we say about the distribution of $Y$?
I guess the moments are easy to deduce?
Thanks in advance
 A: The first moment (i.e. the mean) is quite easily done, because it's just a constant times the second factorial moment of the original Poisson, which has very simple form and is easy to derive.
That is, you can calculate $E[X(X-1)]$ very easily and so $E[Y]$ is straightforward.
The variance of $Y$ can be obtained in a couple of ways, though there's something to be said for considering $E[Y(Y-2)]$ (which we might call the second double-factorial moment by analogy with the double factorial) - or something similar, such as $E[(Y+1)(Y-1)]$ - and adjusting that to get the variance (but you must take care if you take that approach, it's easy to leave something out). I'd suggest you at least check your answer by simulation (largish simulations at a variety of values of $\lambda$ helped me see when I had made mistakes, by showing my first attempts at answers couldn't be correct)
Higher moments can be done but are more effort.
It might be possible to do something with the mgf but I haven't attempted it.
All the quantiles follow directly from the relationship to the Poisson, as does the mode.
Your pmf is on the right lines but can be set up a bit more formally. 
