hmwk9sol2027 - Solutions to Homework 9, ISyE 2027 Fall 2006...

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Problem 1 (a) L is a uniform random variable on the interval [0 , 10], so E [ L ] = Z 10 0 t (1 / 10) dt = 5 . (b) Clearly, E [ L 2 ] = Z 10 0 t 2 (1 / 10) dt = 100 / 3 so we see that the variance of L is just V ar ( L ) = 100 / 3 - 100 / 4 = 100 / 12 = 25 / 3 . (c) The c.d.f. of L is the function F ( t ), for t ∈ < (this must be computed for all t ). Notice that when t < 0, F ( t ) = Z t -∞ f ( s ) ds = 0 . When 0 t < 1, F ( t ) = Z t -∞ f ( s ) ds = Z t 0 (1 / 10) ds = t/ 10 . , Finally, when t 1, F ( t ) = Z t -∞ f ( s ) ds = Z 1 0 (1 / 10) ds = 1 . Putting everything together, we see that the cdf of L is just F ( t ) = 0 , t < 0; t/ 10 , 0 t < 10; 1 , t 10. (d) The round trip travel time R consists of walking to the item, picking up the item, and walking back to the initial point. Therefore, R = L/ 2+3+ L/ 2 = L +3 (e) The mean of R is just E [ R ] = E [ L + 3] = 8. (f) The variance of R is just V ar ( R ) = V ar ( L ) = 25 / 3. (g) The c.d.f. of R (which we will denote as F R ) is just F R ( t ) = 0 , t < 3; ( t - 3) / 10 , 3 t < 13; 1 , t 13. 1
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hmwk9sol2027 - Solutions to Homework 9, ISyE 2027 Fall 2006...

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