# 5.4 Differential entropy

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In this module we consider differential entropy.

Consider the entropy of continuous random variables. Whereas the (normal) entropy is the entropy of a discrete random variable, the differential entropy is the entropy of a continuous random variable.

## Differential entropy

Differential entropy
The differential entropy $h(X)$ of a continuous random variable $X$ with a pdf $f(x)$ is defined as
$h(X)=-\int_{()} \,d x$ f x f x
Usually the logarithm is taken to be base 2, so that the unit of the differential entropy is bits/symbol. Note that is the discrete case, $h(X)$ depends only on the pdf of $X$ . Finally, we note that the differential entropy is the expected value of $-\lg f(x)$ , i.e.,
$h(X)=-E(\lg f(x))$

Now, consider a calculating the differential entropy of some random variables.

Consider a uniformly distributed random variable $X$ from $c$ to $c+\Delta$ . Then its density is $\frac{1}{\Delta }$ from $c$ to $c+\Delta$ , and zero otherwise.

We can then find its differential entropy as follows,

$h(X)=-\int_{c}^{c+\Delta } \frac{1}{\Delta }\lg \left(\frac{1}{\Delta }\right)\,d x=\lg \Delta$
Note that by making $\Delta$ arbitrarily small, the differential entropy can be made arbitrarily negative, while taking $\Delta$ arbitrarily large, the differential entropy becomes arbitrarily positive.

Consider a normal distributed random variable $X$ , with mean $m$ and variance $\sigma ^{2}$ . Then its density is $\sqrt{\frac{1}{2\pi \sigma ^{2}}}e^{-\left(\frac{(x-m)^{2}}{2\sigma ^{2}}\right)}$ .

We can then find its differential entropy as follows, first calculate $-\lg f(x)$ :

$-\lg f(x)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\lg e()\frac{(x-m)^{2}}{2\sigma ^{2}}$
Then since $E((X-m)^{2})=\sigma ^{2}$ , we have
$h(X)=\frac{1}{2}\lg (2\pi \sigma ^{2})+\frac{1}{2}\lg e=\frac{1}{2}\lg (2\pi \times e\sigma ^{2})$

## Properties of the differential entropy

In the section we list some properties of the differential entropy.

• The differential entropy can be negative
• $h(X+c)=h(X)$ , that is translation does not change the differential entropy.
• $h(aX)=h(X)+\lg \left|a\right|$ , that is scaling does change the differential entropy.
The first property is seen from both and . The two latter can be shown by using .

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capital labour entrepreneur natural resources
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types of demand and the explanation
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other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand
Mian
if the demand increase supply also increases
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you are wrong this is the law of demand and not the definition
Tarasum
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founder , that is Adam Smith
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Yes very sure it was released in 1759
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thank you Yusuf.
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then when did he died?
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17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
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1790
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monopsony is when there's only one buyer while monopoly is when there's only one producer.
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(uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
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wealth on nation, 1776
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three branches of economics in which tourism is likely to figure
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