Question

Poisson distribution

The Environmental Protection Agency (EPA) establishes the standards to guarantee the quality of air emissions by companies. The maximum allowed limit of copper in the emissions is 10 particles per million and you work in a company where the average value in your emissions is four particles per million.

a) If X is defined as the number of particles per million in a sample, what is the standard deviation of X in your company?

b) If the average number of particles per million in your company is effectively four per million, would you be afraid that the agency will fine you for polluting the air? Justify your answer.

Answer 1

(a)

For a Poisson distribution the numerical value of the mean and the variance is same.

Thus, Variance is 4. It follows then that the standard deviation is 2.

(b)

The probability of X being greater than a particular value away from the mean in a poisson distribution decreases very sharply as we go away from the mean.

Even if you calculate P(X greater than or equal to 10) for this case it will come out as 0.008. This is a very small probability and occurence if it is very low in practical sense.

So, I should not be afraid of agency firing my company