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Question: Let X have a Poisson distribution with mean 16. Estimate P (X≥28) using the normal approximation.

I know that the normal distribution uses the mean and variance. In this case the standard deviation would be √16 which is ~ 4.

Do I just use Z=(x-μ)/σ to find the normal distribution? Which would come to be (28-16)/4 = 3. But that gives me Z and I'm not sure where to go from there.

For this same problem I also had to estimate using Chebyshev's inequality and came up with 1/9 so I'm assuming the normal distribution would give an estimate close to that. I'm just not sure how.

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  • $\begingroup$ Yes, the approximation is given by $Pr(Z\ge 3)$, which is about $0.0013$ I think. Maybe one should use continuity correction, I am not sure it would give an improved approximation in this case. Note this is far from the Chebyshev estimate, which is usually quite poor. $\endgroup$ – André Nicolas Nov 28 '15 at 23:36
  • $\begingroup$ Is there any way to solve for Pr(Z≥3) other than looking at the normal table? Or is that going beyond what the question is asking for? Also, if I wanted to use the continuity correction, how would I do so? $\endgroup$ – Andi Nov 28 '15 at 23:43
  • $\begingroup$ For continuity correction, $\Pr(Z\ge \frac{27.5-16}{4})$. But I am really uncomfortable even using the normal approximation, it is close to the unreliability edge. The numerical error will not be large, but the percentage error could be. As to tables, well, nowadays there are many software solutions, every spreadsheet program does it, some calculators do it. A number of programs, such as R, will directly compute Poisson cdfs. $\endgroup$ – André Nicolas Nov 28 '15 at 23:55
  • $\begingroup$ @Andre What is the rule of thumb as to when/where normal approximation to the Poisson can be used? $\endgroup$ – A.S. Nov 29 '15 at 0:06
  • $\begingroup$ There are various rules of thumb. For the "fat" part $\lambda=15$ is plenty good enough, and one can, with continuity correction, go down to $10$ or even lower. $\endgroup$ – André Nicolas Nov 29 '15 at 0:28

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