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The discrete random variable X is a count variable of 45 Bernoulli trials and can take on values from 0 to 45. The mean is 2.34 and the variance is 18.39. The average probability of scoring 1 in a single Bernoulli trial is $p=2.34/45=0.05$. According to a rule of thumb, I could consider this binomial distribution approximating a Poisson distribution since p is small.

However, it seems that the assumption for a Poisson distribution, that the single Bernoulli trials are independent, does not hold, resulting in overdispersion (the variance is greater than the mean). For this reason I assume my variable approximates either a negative binomial distribution or an extra binomial (=beta binomial) distribution.

edit: it seems that the assumption is not that the 45 trials within one individual are independent, but that the 45 trials of one person is independent of the 45 trials in another person (?)

My question now is, how could I determine which of these two theoretical distributions (extraBin vs NegBin) best approximates my empirical distribution?

Count distrbution:

enter image description here

Proportion (count/45) distribution: enter image description here

If any more information is needed to answer this question I gladly supply this.

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    $\begingroup$ Could you write out the calculation that you used in computing the variance value $18.39$ for a binomial random variable with parameters $(45,0.05)$? The usual formula $np(1-p)$ necessarily gives a variance smaller than $np$, the mean of the binomial random variable. $\endgroup$ – Dilip Sarwate Apr 9 '13 at 13:44
  • $\begingroup$ More apt for stats.stackexchange.com $\endgroup$ – leonbloy Apr 9 '13 at 13:45
  • $\begingroup$ @DilipSarwate I believe the software calculated with an assumption of normal distribution. I will look into my mistake there. @ leonbloy I asked it at stats but I did not get a conclusive response. So I believe more in depth math knowledge is required. $\endgroup$ – Marloes Apr 9 '13 at 13:49
  • $\begingroup$ Try fitting a geometric distribution to the data. $\endgroup$ – Dilip Sarwate Apr 9 '13 at 13:52
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    $\begingroup$ A NegBin with one success is a geometric distribution $\endgroup$ – Dale M Apr 10 '13 at 2:00

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