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I have a doubt regarding when to approximate binomial distribution with Poisson distribution and when to do the same with Normal distribution. It is my understanding that, when p is close to 0.5, that is binomial is fairly symmetric, then Normal approximation gives a good answer. However, when p is very small (close to 0) or very large (close to 1), then the Poisson distribution best approximates the Binomial distribution. Also, when n is large enough to compensate, normal will work as a good approximation even when n is not close to 0.5 (n will work fine, but still Poisson will be better? ) However,consider the following question-

The probability of any given policy in a portfolio of term assurance policies lapsing
before it expires is considered to be 0.15.
For a group of 100 such policies, calculate the approximate probability that more than
20 will lapse before they expire.

Here n is 100 and p is 0.15 (which is not close to 0.5). In this case, the exact answer is 0.0663. The normal approximated answer is 0.06178 and the Poisson approximated answer is 0.08297.

My doubt is that, since p is closer to zero than it is to 0.5, shouldn't the Poisson approximation yield a better answer?

enter image description here

enter image description here

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    $\begingroup$ Why don't you graph the binomial distribution and superpose the normal, and Poisson approximations, and draw a line at $x=20$? $\endgroup$ – Neil G Feb 29 '16 at 4:45
  • $\begingroup$ @NeilG Although, I am studying for an exam where I won't have access to CAS, I still did exactly what you said using Mathematica just for my own understanding. However, the values are so close (only differ after second decimal place), that all the probabilities appear to be exactly the same. Also, that doesn't explain WHY any approximation is better than the other, it just shows us which one is. $\endgroup$ – Utkarsh Mittal Feb 29 '16 at 4:50
  • $\begingroup$ Those don't like like Poisson and Normal distributions, which are continuous. Also, please graph from 0 to at least 50. $\endgroup$ – Neil G Feb 29 '16 at 5:02
  • $\begingroup$ @NeilG Poisson is not continuous. In normal, I have accounted for continuity correction. For example x<=20 implies x<20.5. Also, since the Original Random Variable can only have integral values, Normal probability of say 20.4 is meaningless and equal to saying Original Random variable can take all values upto 20, that is all values upto 20.5 after continuity correction. $\endgroup$ – Utkarsh Mittal Feb 29 '16 at 5:07
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    $\begingroup$ @Glen_b I just figured that the skewness of the Poisson approximation would be more visible with the pdf and that this particular question might not need that aspect of the binomial to be modeled since 20/100 is roughly 0.15? I don't know. $\endgroup$ – Neil G Feb 29 '16 at 8:43
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Here is a pmf plot I was able to create in MATLAB---looks like the normal (Gaussian) is pretty close, where as the Poisson misses the peak and has a fatter long tail.

enter image description here

Furthermore, looking at wiki (not always infallible!), according to NIST/SEMATECH, "6.3.3.1. Counts Control Charts", e-Handbook of Statistical Methods., Poisson is a good approximation for $p < 0.05$ (not 0.5) for $n > 20$.

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