entropyinformation theoryrandom variablessoft-question
Temperature, energy, entropy, etc. are quantities in statistical mechanics, and are properties of specific physical systems.
However, recently I discovered that random variables have "entropy" $S$:
$\text{S}(X) = \mathbb{E}\left[ – \log p(X) \right] = -\sum_x p(x) \log(p(x)) $
This makes me wonder, do random variables/distributions also have defined terms like temperature and energy?
They usually hold the relation: $\dfrac{\partial S}{\partial U} = \dfrac 1T$
Yes, Information Theory is a branch of mathematics, although its practitioners are often found in departments of Electrical and Computer Engineering or Computer Science.
One suggestion is to work with copulas. In a nutshell, a copula allows you to separate out the dependency structure of a distribution function. Say, $F_1,F_2,\ldots,F_n$ are the 1D marginals of a distribution $F$ then the copula $C$ is the function defined as
$$C(u_1,u_2,\ldots,u_n)=F(F^{-1}_1(u_1),F^{-1}_1(u_2),\ldots,F^{-1}_n(u_n))$$
This makes $C$ a function from $[0,1]^n$ to $[0,1]$. For instance, if you take the bivariate normal distribution, by doing the computation above, you'll find the Gaussian copula
$$C^{\text{Gauss}}_{\rho}=\int_{-\infty}^{\phi^{-1}(u_1)}\int_{-\infty}^{\phi^{-1}(u_2)}\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(-\frac{s_1^2-2\rho s_1s_2+s_2^2}{2(1-\rho^2)}\right)ds_1ds_2$$
I used the package copula in R to illustrate. If you just take the copula as such, it is as if you constructed a probability distribution with the dependency structure of a bivariate normal, but with uniform marginals. So I generated 1000 random vectors from a Gaussian copula with $\rho=0.5$. Here's the code
library(copula);
norm.cop <- normalCopula(0.5); u <- rCopula(1000, norm.cop);
plot(u,col='blue',main='Random variables, uniform marginals, gaussian copula, > rho=0.5',xlab='X1',ylab='X2')
cor(u);
and the result 
I also computed the sample correlation which is $0.5060224$.
I also computed a plot to show you the marginals are indeed uniform
dom<-(1:length(u[,1]))/length(u[,1]);
par(mfrow=c(1,2));
plot(dom,sort(u[,1]),col='blue',main='marginal X1'); abline(0,1,col='red');
plot(dom,sort(u[,2]),col='blue',main='marginal X2'); abline(0,1,col='red');
This is all very nice, but there are a number of pitfalls that have to be discussed:
More can be said and I think the article I quoted in my last item is a nice starting point.
Best Answer
Entropy in one branch of mathematics, information theory, is a measure of the information content of a message. It's not actually related to the entropy in physics except by analogy. A completely ordered message consisting of just ones or just zeros has no entropy, whereas a totally random unpredictable message has high entropy.
As far as I know, there is no equivalent use of the words "temperature" or "energy" in information theory, for example the "temperature" of an information source or the amount of "energy" in a message.
Even if these terms were used, they would be analogies to physics concepts rather than meaningful terms.