Question

6. A transmitter randomly sends one of the messages in {ai,a2,.., ,an). The receiver either receives the transmitted message with probability p, or mistakenly receives one of the other messages with equal probabilities, (a) What is the probability of receiving ai at the receiver? (b) If a is received at the receiver, what is the probability that the transmitted message was 1

Answer 1

Question 6

Here let say the transmitter has send the message $\alpha _t$ so here

Pr(Receiving $\alpha _t$) = p

Pr(Receiving any other particular message) = (1 - p)/(n-1)

(a) So here probability of reciving $\alpha _1$ at the receiver is to be evaluated.

Any message can be send with probability 1/n

so here we will receive $\alpha _1$ when transmitted message is $\alpha _1$ and we received the correct message or we were transmitted any message other than $\alpha _1$ and we received $\alpha _1$.

Pr(recieiving $\alpha _1$) = Pr(Transmitted $\alpha _1$) * Pr(Received $\alpha _1$) + Pr(Transmitted other than $\alpha _1$) * Pr(Recieved $\alpha _1$)

= 1/n * p + (n-1)/n * (1-p)/(n-1)

= p/n + (1-p)/n = 1/n

(b) Now we received $\alpha _1$ at the reciever so

here

Probability that transmitted message is $\alpha _1$ = (p/n)/(1/n) = p