6 A sensor network consists of a large number of microproces

6. A sensor network consists of a large number of microprocessors spread over a large area, in communication with a base station. In this network, the probability that a message from a sensor will fail to reach the base station is 0.005. Assuming that during a particular day 1000 messages are sent, (a) What is the probability that exactly three of these messages fail to reach the base station? (b) What is the probability that more than four of these messages fail to reach the base station?

Solution

This is poisson distribution

a.) mean = lambda = np = (1000)(0.005) = 5

P(x) = e^(-lambda)*(lambda)^x/x!

P(exactly 3) = e^(-5)*(5)^3/3! = 0.14037

b.) P(X>4) = 1 - P(X<=3)

= 1 - [P(0) + P(1) + P(2) + P(3)]

= 1 - e^(-5)[ 1 + 5 + 5^2/2 + 5^3/6]

= 0.735