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问题 1.


Exercise 1
(a) Prove that the information measure is additive: that the information gained from observing the combination of $N$ independent events, whose probabilities are $p_i$ for $i=1 \ldots . . N$, is the sum of the information gained from observing each one of these events separately and in any order.

.


Solution:
(a) The information measure assigns $\log 2(p)$ bits to the observation of an event whose probability is $p$. The joint probability of a combination of $N$ independent events whose probabilities are $p_1 \ldots p_N$ is $\prod{i=1}^N p_i$. Thus the information content of such a combination is:
$$
\log 2\left(\prod{i=1}^N p_i\right)=\log _2\left(p_1\right)+\log _2\left(p_2\right)+\cdots+\log _2\left(p_N\right)
$$
which is the sum of the information content of all of the separate events.
(b) Calculate the entropy in bits for each of the following random variables:
(i) Pixel values in an image whose possible grey values are all the integers from 0 to 255 with uniform probability.
(ii) Humans classified according to whether they are, or are not, mammals.
(iii) Gender in a tri-sexed insect population whose three genders occur with probabilities $1 / 4,1 / 4$, and $1 / 2$.
(iv) A population of persons classified by whether they are older, or not older, than the population’s median age.

问题 2.

(b) Calculate the entropy in bits for each of the following random variables:
(i) Pixel values in an image whose possible grey values are all the integers from 0 to 255 with uniform probability.
(ii) Humans classified according to whether they are, or are not, mammals.
(iii) Gender in a tri-sexed insect population whose three genders occur with probabilities $1 / 4,1 / 4$, and $1 / 2$.
(iv) A population of persons classified by whether they are older, or not older, than the

Solution:
(b) By definition, $H=-\sum_i p_i \log _2 p_i$ is the entropy in bits for a discrete random variable distributed over states whose probabilities are $p_i$. So:
(i) In this case each $p_i=1 / 256$ and the ensemble entropy summation extends over 256 such equiprobable grey values, so $H=-(256)(1 / 256)(-8)=8$ bits.
(ii) Since all humans are in this category (humans $\subset$ mammals), there is no uncertainty about this classification and hence the entropy is 0 bits.
(iii) The entropy of this distribution is $-(1 / 4)(-2)-(1 / 4)(-2)-(1 / 2)(-1)=1.5$ bits.
(iv) By the definition of median, both classes have probability 0.5 , so the entropy is 1 bit.

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