Kyung Hee University
Department of Electronic Engineering
Probability and Random Variables
Fall 2017: Exam I
Hyundong Shin
October 12, 2017
16:30 – 17:50 (80 minutes)
Chapters 1–3 (120 points): Show all relevant work.
Problem 1 (50 points)
Suppose that binary symbols are transmitted over the channel X→Yas shown in the following
figure where each transmission is statistically independent.
Note that P[X= 0] = P[X= 1] = 1/2and each label on the arrow denotes the conditional
probability. For example,
=P[Y= 1|X= 0] = P[Y= 0|X= 1] .
Use (x, y)to denote
(X=x, Y =y).
(a) (10 points) Find the sample space S.
(b) (10 points) Find the probability of erasure P[Y=E]and the probability of error P[X6=Y, Y 6=E].
(c) (10 points) Find P[X= 1|Y= 0].
(d) (10 points) Transmit 100 bits over the channel. Find the probability of 10 errors and 5 erasures
in this 100 independent trials.
(e) (10 points) Let Zbe the number of transmitted bits until the first successful reception comes
up. Find the probability mass function (PMF) PZ(z)of Z.
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