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
