## Description

Exercise 1.

Show that if A and B are independent events, then the pairs A and Bc

, Ac and B, and Ac and Bc are also

independent.

c 2020 Stanley Chan. All Rights Reserved. 1

Exercise 2.

A binary communication system transmits a signal X that is either a +2 voltage signal or a -2 voltage signal.

A malicious channel reduces the magnitude of the received signal by the number of heads it counts in two

tosses of a coin. Let Y be the resulting signal. Possible values of Y are listed below.

2 Heads 1 Head No Head

X = −2 Y = 0 Y = −1 Y = −2

X = +2 Y = 0 Y = +1 Y = +2

Assume that the probability of having X = +2 and X = −2 is equal.

(a) Find the sample space of Y , and hence the probability of each value of Y .

(b) What are the probabilities P[X = +2 | Y = 1] and P[Y = 1 | X = −2]?

c 2020 Stanley Chan. All Rights Reserved. 2

Exercise 3.

A computer manufacturer uses chips from three sources. Chips from source A, B and C are defective with

probabilities 0.01, 0.003 and 0.008, respectively. The proportions of chips from A, B, C are 0.2, 0.3, 0.5

respectively. If a randomly selected chip is found to be defective, find

(a) the probability that the chips are from A

(b) the probability that the chips are from B

(c) the probability that the chips are from C

c 2020 Stanley Chan. All Rights Reserved. 3

Exercise 4.

Consider the following communication channel. A source transmits a string of binary symbols through a

noisy communication channel. Each symbol is 0 or 1 with probability p and 1 − p, respectively, and is

received incorrectly with probability ε0 and ε1, respectively. Errors in different symbols transmissions are

independent.

Denote S as the source and R as the receiver.

(a) What is the probability that a symbol is correctly received? Hint: Find P[R = 1 ∩ S = 1] and

P[R = 0 ∩ S = 0].

(b) Find the probability of receiving 0110 conditioned on that 0110 was sent, i.e., P[R = 0110 | S = 0110].

(c) In an effort to improve reliability, each symbol is transmitted three times and the received string is

decoded by majority rule. In other words, a 0 (or 1) is transmitted as 000 (or 111, respectively), and

it is decoded at the receiver as a 0 (or 1) if and only if the received three-symbol string contains at

least two 0s (or 1s, respectively). What is the probability that the symbol is correctly decoded, given

that we send a 0?

(d) Suppose that the scheme of part (c) is used. What is the probability that a 0 was sent conditioned on

that the string 100 was received?

(e) Suppose the scheme of part (c) is used, and given that a 0 was sent. For what value of ε0 is there an

improvement in the probability of correct decoding? Assume that ε0 6= 0.

c 2020 Stanley Chan. All Rights Reserved. 4

c 2020 Stanley Chan. All Rights Reserved. 5

Exercise 5.

Two dice are tossed. Let X be the absolute difference in the number of dots facing up.

(a) Find and plot the PMF of X.

(b) Find the probability that X ≤ 2.

(c) Find E[X] and Var[X].

c 2020 Stanley Chan. All Rights Reserved. 6

Exercise 6.

Let X be a random variable with PMF pk = c/2

k

for k = 1, 2, . . ..

(a) Determine the value of c.

(b) Plot the PMF and the CDF.

(c) Find P(X > 3) and P(1 ≤ X ≤ 4).

(d) Find E[X] and Var[X].

c 2020 Stanley Chan. All Rights Reserved. 7