Exercise $1$

Suppose you have a data source that produces a series of characters drawn from a set of distinct symbols. If symbol is produced with probability $,$ the first$-$order entropy of the source is defined by

This value is the number of bits needed per symbol to encode a long message. It is a measure of the informational content of the message and indicates the potential effectiveness of data compression strategies. If $,$ then there is only one symbol present and no information is present. If all symbols have equal probability, then $.$

Compute the first$-$order entropy by finding the number of unique entries in the message, $,($using unique$),$ then counting the occurrences of each symbol and dividing by the length of $.$

Use the following built$-$in data from MATLAB as your vector $.$

load clown

v $=$ X$($:$)$;

$\%$ image$($X$)$

$\%$ colormap$($map$)$

You may uncomment the last two lines if you want to see the image.