Probability: Variance, Mean and Standard Deviation

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