# approximations for discrete distributions

• May 29th 2009, 07:35 PM
calabrone
approximations for discrete distributions
The number of trees in one acre has a Poisson distribution with mean 60. Assuming independence, compute approximately $\displaystyle P(5950 \le X \le 6100)$, where X is the number of trees in 100 acres.

Could someone work this problem out for me or a similar one? I have a few of this type to do. Thank you
• May 29th 2009, 09:33 PM
Random Variable
The sum of n independent Poisson random variables each with parameter $\displaystyle \lambda_{i}$ is itself a Poisson random variable with parameter $\displaystyle \sum^{n}_{i=1} \lambda_{i}$.

So X is Poisson random variable with parameter 60*n = 60*100 = 6000. Therefore, X has a mean a 6000 and a variance of 6000.

Since n is fairly large, $\displaystyle \frac {X-6000}{\sqrt{6000}}$ is approximately $\displaystyle N(0,1)$ by the central limit theorem

so find $\displaystyle P(\frac {5950-6000}{\sqrt{6000}} \le Z \le \frac {6100-6000}{\sqrt{6000}})$

If you were dealing with a strict inequality, the answer would be different.