Answer :
Calculating the probability of a certain number of messages arriving at a website described by a Poisson process involves summing probabilities from the Poisson distribution formula. Part (a) concerns at most 500 messages in five seconds, while Part (b) addresses the likelihood of more than 125 messages in one second, with each respective average rate calculated from 6000 messages per minute.
The arrival of messages at a Web site can be described using a Poisson process, where the average rate is given as 6000 messages per minute. To answer these questions, we use the properties of the Poisson distribution.
The average number of messages in five seconds (which is 1/12 of a minute) is 6000/12 = 500 messages. Using the Poisson distribution probability mass function formula:
P(X = k) = (λ^k * e^(-λ)) / k!
We sum up the probabilities from X = 0 to X = 500 to find the probability of getting at most 500 messages in five seconds.
To find the probability that more than 125 messages arrive in one second (which is 1/60 of a minute), we first determine the average number of messages in one second, which is 6000/60 = 100 messages. Then we calculate the probability for receiving more than 125 messages in one second using the complement of the sum of probabilities from X = 0 to X = 125.
The actual probabilities require the use of a calculator or software capable of computing these sums for Poisson distribution.
