We are installing a router for our network.

We believe that the time between the arrival of packets will be exponentially distributed with parameter R = 2 packets/second, and that these times are independent given R. We believe that each packet, independently, will either go to the server (with probability S = 0.4), the laser printer (with prob. 0.2), or to one of computers (with prob. 0.4).

We believe that each packet's size in kilobytes, independently, has a Normal(1, 0.01) distribution.

1. A tasty byte

a) Consider the total size of the next ten packets in kilobytes. Find its distribution, expected value and variance, and the probability it is less than 9.5 kilobytes.

b) Consider the largest and the smallest of the next ten packets. Find their distributions and, for each, the probability it is less than 1.01 kilobytes.

c) Consider how many of the next ten packets are less than 1.02 kilobytes. Find its distribution, expected value and variance.

d) Consider how many of the next thousand packets are less than 1.02 kilobytes. Find its expected value and variance, and approximate its distribution and the probability that at least 575 of those packets are less than 1.02 kilobytes.

e) Consider the next three packets to arrive. Find the probability that the first is at least 0.1 kilobytes larger than the average of the other two.