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what is the difference between Poisson distribution as an approximation of Binomial distribution and Normal (Gaussian) distribution as an approximation of Binomial distribution? Both are approximations as number of trials tend to infinity.. then what is really the difference?

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You are right If n tends to large in binomial will tend to either normal distribution or Poisson. The Difference is in the Value of p . If p is close to 1/2 it will tend Normal and if p is very small and np < 5 or np <10 then it will tend to poison.

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    $\begingroup$ (-1) This answer is misleading because the two limiting processes are very different. If $np\lt 10$ as $n\to\infty,$ then obviously $p=0$ and the distribution is constant. Otherwise, as $n\to\infty$ (and $p\ne 1$) the standardized distribution is always Normal. The Poisson limit is attained by varying $p$ along with $n$ so that $np$ approaches a constant. $\endgroup$ – whuber Feb 26 at 14:01

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