(a) Suppose X Pois(). Show that,

E[tx] = e(t1)

(b) Eng, Karol, and Yeh ("A Growable Packet (ATM) Switch Architecture: Design Principles and Applications," (IEEE Transactions on Communications), 1992) describe a data packet switch that uses the statistical behaviour of packet arrivals to reduce interconnection complexity. The arrival of a packet at any output in the system is independent of each other. Suppose 16 packets are in the system and the probability of any one packet arriving at output X is = 0.15

What is the probability that exactly 2 of the packets arrive at output X?

What is the probability at least 2 of the packets arrive at output X?

What is the probability no more than 1 packet arrives at output X?

What is the expected number of packets arriving at output X?

What is the standard deviation of the number of packets arriving at

output X?

(c) A paint factory uses "agent A" in the paint manufacturing process. There is an average of 3 particles of agent A in a cubic foot of the air emitted during the production process. What is the probability that there will be,