In a communication system a zero or one is transmitted with p[X=0]= α =0.2 and p[X=1]=1 - α =0.8, respectively. Due to the noise in the channel, a zero can be received as a one with probability β1=0.25 and a one can be received as a zero with probability β2=0.2. With a probability β3=0.2 either of the transmitted bit is erased.

Assume that a new 'bit' (a zero or a one) is transmitted 2 times successively, independent of one another. Let X be a random variable such that

X = # of erroneous bits received + 2.(# of erasure bits)

Hint: The random variable X can be thought of a sum of two random variables, such as

X = A + 2B

where A and B are random variables such that

A = # of erroneous bits received

B = # of erasure bits.

On the average, find the probability of receiving a bit in error?

On the average, find the probability of having a transmitted bit erased at the receiver?

Find p(A=0)

Find p(A=1)

Find p(A=2)

Find p(B=0)

Find p(B=1)

Find p(B=2)

Find px(3)