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# Probability: Variance, Mean and Standard Deviation

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1) Suppose we have an aisle with storage racks on both sides of the aisle. The aisle is 100 feet long. A worker is stationed at one end of the aisle. The worker needs to retrieve an item from storage. Assume that the items are divided into two groups: high turnover and low turnover. The high turnover items are stored in the locations closest to the workers location. Assume that 10% of the items are designated as high turnover, and they are responsible for 90% of the retrievals. Let L be the distance that the worker needs to walk along the aisle to reach the retrieval location.
a) What would be a reasonable density function for L?
b) What is the average distance that the worker needs to walk to the retrieval location (including units)?
c) What is the variance of L (including units)?
d) What is the standard deviation of L (including units)?

2) In the previous problem, suppose the worker walks at 5 feet per second and takes 10 seconds to remove the item from the storage rack. Let T denote the total time, round trip travel including the time to remove the item from the storage rack.
a) Derive an equation for T in terms of L.
b) Compute the mean of T.
c) Compute the variance of T.
d) Determine the probability density function of T.

Formulas:
Var[X] = E[X^2] - (E[X])^2

Mean of X: E(X)= Integral of xf(x) dx

##### Solution Summary

In about 230 words, this solution demonstrates how to compute the statistical quantities for these probability based problems, asking for values such as the mean and variance. All required calculations and formulas are included in proper format.

##### Solution Preview

Assuming the worker has to walk all the distance of high turnover item or low turnover items to retrieve.

a) Obviously, the high turnover items will cover the first 10 feet of the racks, and the low turnover is the next 90 feet.
The probability of retrieving a high turnover item is 0.9, and the worker will walk 10 feet to retrieve.
The ...

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