Question

question:

Bits 0 and 1 are transmitted in the data transmission channel,  due to a noise, a single bit is incorrectly received by a probability of 0.3 , Playback For encoding instead of bit 0, the code word 00000 and bit 1 are transmitted as code word 11111,The receiver decodes the codeword received by selecting the most frequently occurring bit ,  for example 00000 → 0, 01010 →

(a) What is the probability that the received code word is correctly decoded?

(b) When millions of bits are transmitted over the data channel over the above repetition coding, what is the expected value of decoded codewords incorrectly?

Answer 1

(a) What is the probability that the received code word is correctly decoded?

Solution : -

Let the random variable X represent the number of bits out of the given 5 bits which are transmitted correctly. X is a binomial random variable with N = 5 and p = 1- 0.3 = 0.7

For a code word to be correctly decoded, atleast 3 bits should be decoded correctly. The probability of this event would be given by,

(b) When millions of bits are transmitted over the data channel over the above repetition coding, what is the expected value of decoded codewords incorrectly?

Solution :- For each 5 bits sent there is one codeword. Hence when a million bits are transmitted there are 200,000 codewords. The probability of a single codeword being incorrect would be the complementary probability of what we have calculated in the previous part.

p = 1 - 0.8369 = 0.1631

Hence the expected number of codewords decoded incorrectly would be given by,

E = 0.1631*200,000 = 32,620