If k random variables are independent Poisson distributed random variables with

Question

If k random variables are independent Poisson distributed random variables with parameter A, then the sum of all these random variables has a Poisson distribution with parameter k. Say that 4 independent random variables were identically distributed as Poisson with parameter 10. Let ï be the sample mean of these 4 random variables. What is the exact probability that the interval 19610 +1.96 10/4 contains 10? As a hint, you can use the function ppois(V,40) in R to evaluate the cumulative distri- bution of the Poisson distribution function, which gives the probability a Poisson distributed random variable with = 40 is less than some number

Explanation / Answer

this is 95% confidence interval if it can be approximated by normal distribution

mean = 40

10 + 1.96 * sqrt(10/4) = 13.09903

10 - 1.96 * sqrt(10/4) = 6.90096

ppois(13.099,10)
[1] 0.8644644
> ppois(6.90096,10)
[1] 0.1301414

hence

0.8644644 - 0.1301414 = 0.734323

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