The arrivals of new messages at a computer server is a Poisson process () with an arrival rate of = 4 messages per second. An experiment consists of monitoring the server and recording () over a 10-second interval.

(a) What is the probability of exactly four messages arriving in the first second of observation?

(b) Given that six messages arrived during the first two seconds, what is the conditional probability that two messages arrived during the first second?

(c) Let T denote the inter-arrival time between two messages. Write down the probability density function of the random variable T.

(d) Based on the answer in part (c), find the probability that the inter-arrival time is less than 3 seconds.