2008Midterm2sln

# 2008Midterm2sln - X i< Y 1 ∀ i | Y 1 = y Pr Y 1 = y dy...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Midterm 2 Suggested Solution 1. (a) States: number of tosses since the last head appears. Transition probabilities are characterized as follows: P i, 0 = p i = 0 , 1 , 2 ... P i,i +1 = q = 1 - p i = 0 , 1 , 2 ... (b) π 0 = p i =0 · π i π 1 = q · π 0 · · · π i = q · π i +1 i =0 π i = 1 (c) π 0 = p and π i = q i · π 0 = q i · p 2. Conditioned on the fact that 1 man arrived within the 1 hour period, the distribution of Y 1 (first man’s arrival time) is Uniform[0,1]. Let X i denote the ith women’ arrival time, thus X i s are also distributed as U[0,1]. Pr { X i < Y 1 , i=1,2,3 } = 1
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Unformatted text preview: { X i < Y 1 , ∀ i | Y 1 = y } Pr { Y 1 = y } dy = Z 1 y =0 y 3 dy = 1 4 3. (a) States: 0 (expected revenue \$600) and 1 (expected revenue \$200. The transition probability is as follows: P = ± . 7 0 . 3 . 5 0 . 5 ² (b) π = 0 . 7 · π + 0 . 5 · π 1 π 1 = 0 . 3 · π + 0 . 5 · π 1 π + π 1 = 1 ⇒ π = 5 8 , π 1 = 3 8 (c) E (Revenue) = π · \$600 + π 1 · \$200 = 5 8 · \$600 + 3 8 · \$200 = \$450 > \$400 Yes, it is proﬁtable to open the store. 1...
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