In a digital communication system, the received signal is sampled at a fixed bit rate and sent to a decision device. The noiseless sampled signal X is either +5 or -5 volts which corresponds to data "1" or "0" respectively. However, in the presence of additive noise, the sampled received signal Y consists of 2 components, Y = X +N, where the noise component N is a zero mean normal distribution random variable with a standard deviation of 2.149. The decision device outputs a "1" if Y > 0, else it outputs a "0".

(i) Given that a "1" was transmitted, calculate the probability that the decision device outputs a "1".

(ii) Given that the probability of transmitting a "1" is 0.7, determine the probability that the decision device makes a wrong decision.

(ii) The decision device outputs data bits "1" or "0" at a fixed rate and the data bits are statistically independent. Let p be the probability of producing a "1" and k be the number of "1" out of n bits the decision device produced. Show that the probability distribution of k is Poisson as n approaches infinity. Note that the Eulers number e is given by e = lim(1+1/m)^m