As mentioned in lectures, entropy is measure of how much information content (“surprise”) is present in a system.

Given a set of N symbols, and the probability of each symbol occurring, we can compute the entropy (in bits) as:

$$ S = - \sum^{N}_{i=1}p_i \times \log_2 p_i $$

where $p_i$ is the probability of encountering a given symbol. A worked example of computing entropy was given in lectures. For another example, consider a system with five symbols: $A$; $B$; $C$; $D$; and $E$, each occurring with probabilities: $0.2$; $0.1$; $0.3$; $0.3$; and $0.1$. The entropy of this system is $2.171$ bits, the computation of which is outlined in the following table:

Symbol | $i$ | $p_i$ | $\log_2 p_i$ | $-p_i \times \log_2 \ p_i$ |
---|---|---|---|---|

A | 1 | 0.2 | -2.322 | 0.464 |

B | 2 | 0.1 | -3.322 | 0.332 |

C | 3 | 0.3 | -1.737 | 0.521 |

D | 4 | 0.3 | -1.737 | 0.521 |

E | 5 | 0.1 | -3.322 | 0.332 |

s=2.171 |

Use the table below to help you compute the entropy of a system with five symbols (A, B, C, D, E) with the probabilities 0.0625, 0.25, 0.5, 0.0625 and 0.125 (respectively):

Note: There is an editable worksheet document on Blackboard you can use for your workings in this lab.

Symbol | $i$ | $p_i$ | $\log_2 p_i$ | $-p_i \times \log_2 p_i$ |
---|---|---|---|---|

A | 1 | 0.0625 | ||

B | 2 | 0.25 | ||

C | 3 | 0.5 | ||

D | 4 | 0.0625 | ||

E | 5 | 0.125 | ||

s= ? |