The joint Shannon entropy (in bits) of two discrete random variables 
and 
with images 
and 
is defined as[3]: 16
![{\displaystyle \mathrm {H} (X,Y)=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}P(x,y)\log _{2}[P(x,y)]}](//wikimedia.org/api/rest_v1/media/math/render/svg/1c1b92b469840ecac93f75eff9beef3d03d6bbc1) |
|
(Eq.1) |
where 
and 
are particular values of and , respectively, 
is the joint probability of these values occurring together, and ![{\displaystyle P(x,y)\log _{2}[P(x,y)]}](//wikimedia.org/api/rest_v1/media/math/render/svg/33023e90bccb1921379d4b2368ce7c89ee728254)
is defined to be 0 if 
.
For more than two random variables 
this expands to
![{\displaystyle \mathrm {H} (X_{1},...,X_{n})=-\sum _{x_{1}\in {\mathcal {X}}_{1}}...\sum _{x_{n}\in {\mathcal {X}}_{n}}P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}](//wikimedia.org/api/rest_v1/media/math/render/svg/453bf0f18adcd1b4160557f8e3be47f6202abd1d) |
|
(Eq.2) |
where 
are particular values of , respectively, 
is the probability of these values occurring together, and ![{\displaystyle P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}](//wikimedia.org/api/rest_v1/media/math/render/svg/45d2b2bd535cc3d9c9e463415a4a81bf4eb6194c)
is defined to be 0 if 
.
Definition
The above definition is for discrete random variables and just as valid in the case of continuous random variables. The continuous version of discrete joint entropy is called joint differential (or continuous) entropy. Let and be a continuous random variables with a joint probability density function 
. The differential joint entropy 
is defined as[3]: 249
 |
|
(Eq.3) |
For more than two continuous random variables the definition is generalized to:
 |
|
(Eq.4) |
The integral is taken over the support of 
. It is possible that the integral does not exist in which case we say that the differential entropy is not defined.
Properties
As in the discrete case the joint differential entropy of a set of random variables is smaller or equal than the sum of the entropies of the individual random variables:
[3]: 253
The following chain rule holds for two random variables:
In the case of more than two random variables this generalizes to:[3]: 253
Joint differential entropy is also used in the definition of the mutual information between continuous random variables:
![{\displaystyle \mathrm {H} (X,Y)=-\sum _{x\in {\mathcal {X}}}\sum _{y\in {\mathcal {Y}}}P(x,y)\log _{2}[P(x,y)]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1c1b92b469840ecac93f75eff9beef3d03d6bbc1)
![{\displaystyle P(x,y)\log _{2}[P(x,y)]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/33023e90bccb1921379d4b2368ce7c89ee728254)
![{\displaystyle \mathrm {H} (X_{1},...,X_{n})=-\sum _{x_{1}\in {\mathcal {X}}_{1}}...\sum _{x_{n}\in {\mathcal {X}}_{n}}P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/453bf0f18adcd1b4160557f8e3be47f6202abd1d)
![{\displaystyle P(x_{1},...,x_{n})\log _{2}[P(x_{1},...,x_{n})]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/45d2b2bd535cc3d9c9e463415a4a81bf4eb6194c)
![{\displaystyle \mathrm {H} (X,Y)\geq \max \left[\mathrm {H} (X),\mathrm {H} (Y)\right]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d8ef6202e31c5bf393a72c24d9972de4e24560f9)
