Answer :
a. The total number of requests (online and run) to arrive between 8:00 am and 10:00 am can be found by using the Poisson distribution formula.
The expected number of queries arriving in 2 hours can be calculated as shown below:E(X) = λ * tE(X) = 90 * 2 = 180E(Y) = λ * tE(Y) = 10 * 2 = 20E(X + Y) = E(X) + E(Y)E(X + Y) = 180 + 20E(X + Y) = 200Therefore, the expectation of the total number of requests to arrive between 8:00 am and 10:00 am is 200.b. If 60 queries arrive between 9:00 am and 9:30 am, the number of run requests follows the Poisson distribution with a rate of λ = 10 per hour. The expected number of run requests can be calculated as shown below:E(Y | X = 60) = λ * tE(Y | X = 60) = 10 * 0.5E(Y | X = 60) = 5.
Therefore, the expectation of the number of run requests for that same time period is 5.c. If 25 queries arrive between 10:00 am and 10:30 am, then the total number of queries arriving between 10:00 am and 11:00 am follows a Poisson distribution with a rate of 90 per hour. The expected number of queries can be calculated as shown below:E(X | X = 25) = λ * tE(X | X = 25) = 90 * 1E(X | X = 25) = 90.
Therefore, the expectation of the total number of queries for 10:00 am to 11:00 am is 90 + 25 = 115.d. If 8 run requests arrive between 1:00 pm and 1:30 pm, then the number of requests arriving in the next 10 minutes follows a Poisson distribution with a rate of λ = 10 / 6 per 10 minutes. The probability that no requests will arrive between 1:30 pm and 1:40 pm can be calculated as shown below:P(X = 0) = e^(-λt) * (λt)^x / x!P(X = 0) = e^(-10/6 * 1/6) * (10/6 * 1/6)^0 / 0!P(X = 0) = 0.824.
Therefore, the probability that no requests will arrive between 1:30 pm and 1:40 pm is 0.824.e. If the last run request was 20 minutes ago and the last query was 5 minutes ago, then the probability that no requests (run or query) will occur in the next two minutes can be calculated as shown below:P(X = 0) = e^(-λt) * (λt)^x / x!P(X = 0) = e^(-(90/60 + 10/60) * 2) * ((90/60 + 10/60) * 2)^0 / 0!P(X = 0) = 0.018Therefore, the probability that no requests (run or query) will occur in the next two minutes is 0.018.
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