# Poisson Distribution Question - Please explain with full steps

**Question description**

The number of imperfections in an object has a Poisson distribution with a mean λ = 8.5. If the number of imperfections is 4 or less, the object is called “top quality.” If the number of imperfections is between 5 and 8 inclusive, the object is called “good quality.” If the number of imperfections is between 9 and 12 inclusive, the object is called “normal quality.” If the number of imperfections is 13 or more, the object is called “bad quality.” The number of imperfections in different objects are independent of each other.

(a) A set of seven articles is taken. What is the probability that the set has exactly two top-quality, two good-quality, two normal-quality and one bad-quality objects?

(b) A set of ten articles is taken. What are the expectation and the standard deviation of the number of normal quality objects in the set.

(c) A set of eight articles is taken. What is the probability
that the sum of the number of top quality and good
quality objects is three or less?

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