According to the FDA \"Food Defect Action Levels\" booklet there is an average o

Question

According to the FDA "Food Defect Action Levels" booklet there is an average of 3 insect fragments per 10 grams of peanut butter. Suppose we let X equal the number of insect fragments in a randomly chosen 10 gram serving of peanut butter. It is reasonable that X is a Poisson random variable with a parameter of 3. What is the expected value of X? What is the variance of X? What is the standard deviation of X? What is the variance of 4X? What is the probability that there will be no insect fragments in the serving? What is the probability that there will be exactly 3 insect fragments in the serving? What is the probability that no more than 2 insect fragments are in the serving? What is the probability that more than 2 insect fragments are in the serving? What is the probability that between 3 and 5 (3 lessthan=X lessthan=5) insect fragments are in the serving? We independently select 3 10-gram peanut butter servings. What is the probability that the maximum number of insect fragments in the 3 servings is lessthan = 2?

Explanation / Answer

a)

The expected value is the mean,

E(X) = 3 [ANSWER]

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b)

For Poisson variable,

Var(X) = E(x) = 3 [ANSWER]

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c)

Hence,

sigma(x) = sqrt(Var(x)) = sqrt(3) = 1.732050808 [ANSWER]

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d)

As

Var(4x) = 4^2 Var(x)

then

Var(4x) = 4^2 * 3 = 48 [ANSWER]

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e)

Note that the probability of x successes is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    3      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.049787068 [ANSWER]

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