- #1

Deimantas

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## Homework Statement

The data contains 2500 integers, each is either a 0, 1 or 2:

zeroes: 1240

ones: 1014

two's: 246

Does the data follow Poisson, geometric, binomial or negative-binomial distribution?

## Homework Equations

## The Attempt at a Solution

The mean of the data is

**0.6024**and the variance is

**0.436314**

Negative-binomial distribution is supposed to have greater variance than mean, so I only consider Poisson, binomial and geometric distributions.

Poisson is supposed to have it's mean equal to it's variance. I don't know if I should reject Poisson though, after using method of moments and setting λ=0.6024 I get these theoretical values of distribution:

zeroes:1369

ones:825

twos:248

It's not really that far off. However, Chi-squared test gives me a value of χ

^{2}≈55 which is very large and tells me the hypothesis that my data follows Poisson distribution should be rejected.

I tried generating random Poisson distribution values with λ=0.6024 and got

zeroes:1348

ones:880

twos:214

which gives χ

^{2}≈33. Closer, but still too large.

As for binomial distribution, using method of moments I get 2500*p=0.6024; p=0.000241

With this estimator, using theoretical formulas for calculating binomial probabilities I end up with these values:

zeroes:1369

ones:825

twos:248

These are identical to the theoretical Poisson distribution values. However, when I try to generate 2500 binomial distribution random values with p=0.000241 I get very different results, something like:

zeroes:1850

ones:531

twos:97

I don't really know why it differs.

Finally, geometric distribution. I really did not know what estimators I should use for this one. I tried using (1-p)/p=0.6024 which gives theoretical values of

zeroes:1506

ones:599

twos:238

The randomly generated values were very close to these, but it's quite far off from my data.

So, after all, I still have no clue which distribution does my data follow. Could you help me with that?