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Example Class 6 (sln)

# Example Class 6 (sln) - The University of Hong Kong...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11) Example Class 6 1. Customers arrive at a travel agency at a mean rate of 9 per hour in accordance with a Poisson process. Suppose the travel agency opens at 9:00 am. a. Find the probability that no customers arrived before 9:30 am. Let 𝑋𝑋 be the number of customers arrived from 9:00 to 9:30 pm. Then 𝑋𝑋 ~ 𝑃𝑃𝑃𝑃𝑃𝑃 (4.5) [Since the period is half an hour.] 𝑃𝑃 ( 𝑋𝑋 = 0) = 𝑒𝑒 4.5 = 0.0111 b. Find the probability that more than five customers arrived before 9:30 am. 𝑃𝑃 ( 𝑋𝑋 > 5) = 1 − 𝑃𝑃 ( 𝑋𝑋 ≤ 5) = 1 − � 𝑒𝑒 4.5 4.5 𝑥𝑥 𝑥𝑥 ! 5 𝑥𝑥 =0 = 1 − 𝑒𝑒 4.5 1 + 4.5 + 4.5 2 2! + + 4.5 5 5! = 0.2971 c. Find the probability that there were more than five customers arrived before 9:30 am, given that at least one customers arrived within that period. 𝑃𝑃 ( 𝑋𝑋 > 5| 𝑋𝑋 ≥ 1) = 𝑃𝑃 ( 𝑋𝑋 > 5 ∩ 𝑋𝑋 ≥ 1) 𝑃𝑃 ( 𝑋𝑋 ≥ 1) = 𝑃𝑃 ( 𝑋𝑋 > 5) 1 − 𝑃𝑃 ( 𝑋𝑋 < 1) = 0.2971 1 0.0111 = 0.3004 2. Let 𝑍𝑍 ~ 𝑁𝑁 (0,1) and 𝑋𝑋 ~ 𝑁𝑁 (80,100) . Find the following probabilities, 𝑥𝑥 or 𝑧𝑧 .

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