# If X is Possion($\lambda$), what is the PMF of X / $\lambda$

My idea is by doing a transformation, let $$X' = X / \lambda$$, then $$X = X' \lambda$$

Since $$X \sim Poisson(\lambda)$$, $$\displaystyle P(X'=k) = P(X = \lambda k) = \displaystyle \frac{\lambda ^ {k \lambda}e^{-\lambda}}{(k\lambda)!}$$

Is this correct?

• $X^\prime$ merely uses a different set of numbers to designate the values of $X.$ Changing how you write down an event doesn't change the probabilities, which remain the same they always were.
– whuber
Feb 13, 2021 at 23:05
• What is $(k\lambda)!$ when $k\lambda$ is not an integer? Say $k=2$ and $\lambda=\pi$? Feb 13, 2021 at 23:39
• @Dilip That is usually understood to mean $\Gamma(k\lambda+1).$ (But you likely know that and are trying to suggest that taking the factorial of a non-integral value might be a red flag to detect a possible error.)
– whuber
Feb 14, 2021 at 0:06
• @whuber Yes, many people would take $t!$ to mean $\Gamma(t+1)$ when $t$ is not an integer but I am not sure if the OP is one of the many. His calculations clearly suggest that the possibility that $k\lambda$ might not be an integer had never occurred to him. Feb 14, 2021 at 15:06
• @Dilip I'm not so sure about that. I think we're getting hung up on the idea that "$k$" must refer to an integer. If instead we recognize that implicitly $k$ must be a non-negative integral multiple of $1/\lambda,$ then automatically, $k\lambda$ is an integer and everything works out. Of course it's essential to characterize the possible values of $k$ in any solution.
– whuber
Feb 14, 2021 at 18:59

Comment. Not quite: You need to specify precisely what values the random variable $$X^\prime$$ can take: Consider an example in R with $$\lambda = 5.$$

set.seed(213)
x = rpois(10^5, 5)
table(x)
x
0     1     2     3     4     5     6     7     8
702  3393  8382 14181 17681 17424 14646 10365  6407
9    10    11    12    13    14    15    16    17
3612  1750   856   393   134    52    15     6     1

x.1 = x/5
summary(x.1)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.0000  0.6000  1.0000  0.9984  1.2000  3.4000
table(x.1)
x.1
0   0.2   0.4   0.6   0.8     1   1.2   1.4   1.6
702  3393  8382 14181 17681 17424 14646 10365  6407
1.8     2   2.2   2.4   2.6   2.8     3   3.2   3.4
3612  1750   856   393   134    52    15     6     1