hmwk9sol2027

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online