Question 15 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

## Q15. A source alphabet consists of N symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amount and decreases that of the second by . After encoding, the entropy of the source

## (A) increases

## (B) remains the same

## (C) increases only if N=2

## (D) decreases

## Solution

Entropy of a random variable is defined as ,

.

**Refer **Chapter 2 in Elements of Information Theory, Thomas M. Cover, Joy A. Thomas (Buy from Amazon.com, Buy from Flipkart.com)

Let us consider a simple case where can take two values 1 and 0 with probability and respectively, i.e.

.

The entropy of is,

.

The plot of the entropy versus the probability is shown in the figure below.

clear all; close p = [0:.001:1]; hx = -p.*log2(p) - (1-p).*log2(1-p); plot(p,hx); xlabel('probability, p'); ylabel('H(X)'); title('entropy versus probability, p'); axis([0 1 0 1]);grid on;

**Figure : Entropy versus probability for binary symmetric source**

It can be see that the entropy (also termed as uncertainty) is maximum when and for other values of , the entropy is lower. The entropy becomes 0 when i.e. when the value of becomes deterministic. If we extend this to a source with more than two symbols, **when probability of one of the symbols becomes more higher than the other, the uncertainty decreases and hence entropy also decreases**.

**Based on the above, the right choice is (D) decreases**

** **

## References

[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf

**[2] Elements of Information Theory, Thomas M. Cover, Joy A. Thomas (Buy from Amazon.com, Buy from Flipkart.com)**

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