Question

In: Statistics and Probability

A source transmitted a message through a noisy channel. Each symbol is 0 or 1 with...

A source transmitted a message through a noisy channel. Each symbol is 0 or 1 with probability p and 1 − p, respectively and is received incorrectly with probability 0 and 1. Errors in different symbol transmission are independent.

(a) What is the probability that the kth symbol is received correctly?

(b) What is the probability that the string of symbols 0111 is received correctly?

(c) To improve reliability, each symbol is transmitted three times and the received string is decoded by majority rule. For example, a 0 is transmitted as 000 and is decoded at the receiver as a 0 if and only if the received three symbol string contains at least two 0s. Similar rule applies to 1. What is the probability that a 0 is corrected decoded?

Expert Solution

Answer :

Given that :

Probability of noise communication is p and 1 - p

and let us assume that they received incorrectly with probabilities $\varepsilon o$ and 1 - $\varepsilon o$

a)lets take the values 0 and 1

then the probability that kth symbol will be received correctly as :

P(0 is sent and received) + P(1 is sent and received)
i.e P(1 - $\varepsilon o$ ) + (1 - p)(1 - $\varepsilon 1$ )

b)given that probability that received correctly is 0111

i.e (1 - $\varepsilon 0$ ) * (1 - $\varepsilon 1$ ) * (1 - $\varepsilon 1$ ) * (1 - $\varepsilon 1$ )

= $(1 - \varepsilon 0)$ * $(1 - \varepsilon 1)^3$

The probability that received correctly is 0111 = $(1 - \varepsilon 0)$ * $(1 - \varepsilon 1)^3$

c)Given that the probability that received correctly = 0

i.e P(Received at least 2 0's in string) = 3 $(1 - \varepsilon 0)^2$ * $(1 - \varepsilon 1)$ * $(1 - \varepsilon 0)^3$

orchestra answered 2 years ago

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