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# chapter19 - 1 Let n denote the number of coin flips Then...

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1 Chapter 19 19–1. Let n denote the number of coin flips. Then the number of heads observed is X Bin( n, 0 . 5). Therefore, we can expect to see about n/ 2 heads over the long term. 19–2. If ˆ π n denotes the estimator for π after n darts have been thrown, then it is easy to see that ˆ π n (4 /n )Bin( n, π/ 4). Then E π n ) = π , and we can expect to see the estimator converge towards π as n becomes large. 19–3. By the Law of the Unconscious Statistician, E ( ˆ I n ) = b - a n E n X i =1 f ( a + ( b - a ) U i ) = ( b - a ) E [ f ( a + ( b - a ) U i )] = ( b - a ) Z 1 0 f ( a + ( b - a ) u ) · 1 du = I 19–4. (a) The exact answer is Φ(2) - Φ(0) = 0 . 4772. The n = 1000 result will tend to be closer than the n = 10. (b) We can instead integrate over R 4 0 , say, since R 10 4 0. This strategy will prevent the “waste” of observations on the trivial tail region. (c) The exact answer is 0. 19–5. (a) customer arrival time begin service service time depart time wait 1 3 3.0 6.0 9.0 0.0 2 4 9.0 5.5 14.5 5.0 3 6 14.5 4.0 18.5 8.5 4 7 18.5 1.0 19.5 11.5 5 13 19.5 2.5 22.0 6.5 6 14 22.0 2.0 24.0 8.0 7 20 24.0 2.0 26.0 4.0 8 25 26.0 2.5 28.5 1.0 9 28 28.5 4.0 32.5 0.5 10 30 32.5 2.5 35.0 2.5

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2 Time Event Customers in System 3 Cust 1 arrival 1 4 Cust 2 arrival 1 2 6 Cust 3 arrival 1 2 3 7 Cust 4 arrival 1 2 3 4 9 Cust 1 depart 2 3 4 13 Cust 5 arrival 2 3 4 5 14 Cust 6 arrival 2 3 4 5 6 14.5 Cust 2 depart 3 4 5 6 18.5 Cust 3 depart 4 5 6 19.5 Cust 4 depart 5 6 20 Cust 7 arrival 5 6 7 22 Cust 5 depart 6 7 24 Cust 6 depart 7 25 Cust 8 arrival 7 8 26 Cust 7 depart 8 28 Cust 9 arrival 8 9 28.5 Cust 8 depart 9 30 Cust 10 arrival 9 10 32.5 Cust 9 depart 10 35 Cust 10 depart Thus, the last customer leaves at time 35.
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chapter19 - 1 Let n denote the number of coin flips Then...

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