8 per hour, so that the number of arrivals during a time period of t hours is a

Question

8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter -8t. (Round your answers to three decimal places.) (a) What is the probability that exactly 5 small aircraft arrive during a 1-hour period? What is the probability that at least 5 small aircraft arrive during a 1-hour period? What is the probability that at least 14 small aircraft arrive during a 1-hour period? (b) What is the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? expected value standard deviation (c) What is the probability that at least 20 small aircraft arrive during a 2.5-hour period? What is the probability that at most 16 small aircraft arrive during a 2.5-hour period?

Explanation / Answer

P(x; ) = (e-) (x) / x! Mean Mean 8 20 Poisson rate 8 per hour Exact No. For time t 8t 0 0.000335 2.06115E-09 1 0.002684 4.12231E-08 a) Exactly 5 in 1 hour period 0.002684 2 0.010735 4.12231E-07 3 0.028626 2.7482E-06 Atleast 5 in 1 hour period 4 0.057252 1.3741E-05 For this (1-exact4-exact3-exact2-exact1-exact0) 0.900368 5 0.091604 5.49641E-05 6 0.122138 0.000183214 Atleast 14 in 1 hour period 0.034181 7 0.139587 0.000523468 Same as above 8 0.139587 0.001308669 9 0.124077 0.002908153 b) 90 min duration 10 0.099262 0.005816307 So mean changes to 1.5 times 12 11 0.07219 0.010575103 Expected value is nothing but mean and so 12 12 0.048127 0.017625171 Standard dev is square root of mean and so 3.464102 13 0.029616 0.027115648 14 0.016924 0.03873664 c) 2.5 hour period 15 0.009026 0.051648854 Mean 20 16 0.004513 0.064561067 17 0.002124 0.075954196 Atleast 20 (1-exact19-exact18-……-exact0) 0.529743 18 0.000944 0.084393552 19 0.000397 0.088835317 Atmost 16 (Exact16+exact15+…….+exact0) 0.221074 20 0.000159 0.088835317

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