Categorical Data Question

Hi, I have a problem here that appears to involve Goodness-of-Fit, but my
textbook does not spell out exactly how to solve this case.

I have data from an experiment for a random variable X, where

X Frequency
0 X0
1 X1
2 X2
....


I have a choice of modelling X as a Poisson or Normal distribution.

I'm guessing one should test goodness-of-fit as a Poisson, then choose
normal if it rejects. If I want to test the Poisson distribution, which
estimate of lambda should I use? Should I use the sample mean, or find the
MLE to use variance data as well?

Then how should I test goodness of fit? Should I use chi-square or
G-square?

Thanks



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