Question

Problem 1: Digital Transmission Over Noisy Channels.

Consider transmitting a two digit message over a noisy channel having error probability P(E) = 1/5 per digit. Let X be the random variable that represents the number of possible errors.

(1a). What is the probability that no (zero) error occurs?

(1b). What is the probability that one error occurs?

(1c). What is the probability that more than one error occurs?

(1d). Write the expression for the p.d.f. pX(x) and plot it.

(1e). Write the expression for the c.d.f. FX(x) and plot it.

Let Y be the random variable that represents the number of possible correct digit transmissions.

(1f) Give the expression for random variable Y as a function of random variable X.

(1g). Using the formula for finding p.d.f. of a function (transformation) of random variable and using the p.d.f. of X, write the expression for the p.d.f. pY (y) and plot it.

(1h) Write the expression for the c.d.f. FY (y) and plot it.

Answer 1

(there are more than 4 parts, as per policy i am answering first 4 parts)

problem 1 :

P(E) = 1/5 = 0.2 = p

x = no. of errors

each digit indepedendent therefore binomial distribution

P(X errors) = 2Cx * (p^x) * (1-p)^(2-x)

1a.

P(x=0) = 2C0 * (p^0) * (1-p)^(2-0)

= (1-p)^2 = (1-0.2)^2 = 0.64

P(no error) = 0.64

1b.

P(x=1) = 2C1 * (p^1) * (1-p)^(2-1)

= 2*p*(1-p) = 2*0.2*(1-0.2) = 0.32

P(1 error) = 0.32

1c.

more than one means 2 because only 2 digits are there

P(x=2) = 2C2 * (p^2) * (1-p)^(2-2)

= p^2 = 0.2^2 = 0.04

P(more than 1 error) = 0.04

1d.

pdf for X

each digit indepedendent therefore binomial distribution :

P(X) = 2Cx * (p^x) * (1-p)^(2-x)

P(0) = 0.64

P(1) = 0.32

P(2) = 0.04

plot of pX(x)

(please UPVOTE)