# Statistics - variance and expectation

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It has been determined empiracally that for a certain egg producer, the number of cracked or broken eggs X, in a randomly slected box containing six eggs, follows approximately the distribution given in the table below:

X: 0 1 2 3 4 5 6

P(X): 0.9 0.07 0.02 0.005 0.003 0.001 0.001

find the expectation and variance of X.

if a consignment of eggs from this producer contains 1000 boxes, use the central limit theorem to estimate the probability that the total number of cracked or broken eggs

i) exceeds 175

ii) is at least 150 but does not exceed 190.

indicate any assumptions you have made, and indicate briefly how these might be violated.

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##### Solution Summary

This shows how to use central limit theorem to calculate probability. The variance and expectations in statistics is examined.

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it has been determined empiracally that for a certain egg producer, the number of cracked or broken eggs X, in a randomly slected box containing six eggs, follows approximately the distribution given in the table below:

X: 0 1 2 3 4 5 6

P(X): 0.9 0.07 0.02 0.005 0.003 0.001 0.001

find the expectation and variance of X. ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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