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?
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1$\begingroup$ $X^\prime$ 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. $\endgroup$– whuber ♦Feb 13, 2021 at 23:05
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1$\begingroup$ What is $(k\lambda)!$ when $k\lambda$ is not an integer? Say $k=2$ and $\lambda=\pi$? $\endgroup$– Dilip SarwateFeb 13, 2021 at 23:39
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1$\begingroup$ @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.) $\endgroup$– whuber ♦Feb 14, 2021 at 0:06
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1$\begingroup$ @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. $\endgroup$– Dilip SarwateFeb 14, 2021 at 15:06
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$\begingroup$ @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. $\endgroup$– whuber ♦Feb 14, 2021 at 18:59
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1 Answer
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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
