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Assignment-1 An Introduction to Information Theory

Assignment-1 An Introduction to Information Theory
Assignment-1
Q 1: Which of the following statement is correct?
(a)
H(Y; Z=X) H(Y=X) + H(Z=X)
(b)
H(Y; Z=X) H(Z=X) + H(Y=X)
(c)
H(Y; Z=X) = H(Y=X) + H(Z=X; Y )
(d) All of the above
Q 2: Given a ternary source X = fx1; x2; x3g with probability of occurrence of x1; x2; x3 given by 0.25, 0.25 and 0.5
respectively. What is the information content in
X (in bits)?
(a) 1.5
(b) 0.45
(c) 1.04
(d) 0.9
Q 3: Given H(Y=X) = 0, which of the following statement is correct?
(a)
X = g(y), where g is an arbitrary function
(b)
Y = g(X), where g is an arbitrary function
(c)
I(X; Y ) = H(X)
(d) None of the above
Q 4: Given a discrete random variable X that takes K different values x1; x2; : : : ; xK with probabilities p1; p2; : : : ; pK
respectively. Define a new random variable Y = g(X) where g is an arbitrary function. Which of the statement is
incorrect?
(a)
H(X) = Pi pi log pi
(b) H(X) H(g(X))
(c)
H(g(X)) H(X)
(d) None of the above
Q 5: Two coins are given. One of them is an unbiased coin with equal probability of occurrence of heads and tails
and another is a two headed coin. A coin is selected at random and tossed two times, and number of tails is recorded.
Let
X denote the random variable that has value 0 or 1 depending upon whether unbiased or two headed coin is
chosen. Let
Y denotes the number of tails obtained. (Use the same statement for Q-05 to Q-07)
What is
H(X)?
(a) 0 bit
(b) 0.5 bits
(c) 1 bit
1

(d) 2 bits
Q 6: What is H(Y )?
(a) 0.4 bits
(b) 1.3 bits
(c) 0.9 bits
(d) 1 bit
Q 7: What is I(X; Y )?
(a) 0.38 bits
(b) 0.55 bits
(c) 0.17 bits
(d) None of the above
Q 8: Information theory does not answer following questions.
(a) How to quantify information?
(b) What is the maximum data compression possible for a source?
(c) What is the best coding scheme for error-free transmission over a communication channel?
(d) What is the maximum transmission rate possible over a channel for arbitrarily small error probability?
Q 9: For any two L-ary random variables, X and Y , let p = Pr(X 6= Y ). Which of the following statement is
incorrect?
(a)
H(p) + p log L H(X=Y )
(b)
H(p) + p log(L 1) H(X=Y )
(c) 1 +
p log L H(X=Y )
(d) None of the above
Q 10: Which of the following is a concave function?
(a) 1
log X; X 2 [0; 1)
(b)
I(X; Y ) as a function of p(X) for a fixed p(Y=X)
(c)
I(X; Y ) as a function of p(Y=X) for a fixed p(X)
(d)
D(pjjq) as a function of (p; q)
2

 

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Assignment-1 An Introduction to Information Theory

Assignment-1 An Introduction to Information Theory

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