# Tagged Questions

This tag is for [Entropy](http://en.wikipedia.org/wiki/Entropy_(information_theory)) in Mathematics.

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### Password entropy of famous xkcd comic

The famous xkcd comic about password strength calculates the entropy of the 11-character password "Tr0ub4dor&3" with 28 bits of entropy. When following the ASCII-95-chart, we have 95 possible ...
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### Does entropy inequality hold for convex combination

I have two pairs of Random Variables, $(\mathbb{X},\mathbb{Y})$ and $(\mathbb{M},\mathbb{N})$ which satisfies, $H(\mathbb{X})>H(\mathbb{Y})$ and $H(\mathbb{M})>H(\mathbb{N})$. For some convex ...
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### Proving that the entropy is zero given conditional entropies

Let's suppose we have 4 random variables $X,Y,Z$ and $T$ and that the following equations hold about the entropy: $$H(T|X)=H(T)$$ $$H(T|X,Y)=0$$ $$H(T|Y)=H(T)$$ $$H(Y|Z)=0$$ $$H(T|Z)=0$$ Also, the ...
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### conditional mutual information

I have a question about mutual information $$I(Z ; T/X,Y) = I(T/X,Y ; Z)$$ $T,X,Y,Z$ are random variables is this statement accurate? if it is true and I know that I(Z;T/X,Y) = H(Z/X,Y) - H(Z/T,X,...
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### Entropy and Mutual Information

Consider two discrete random variables $X$ $\{x_1,x_2,\dots,x_n\}$ and $Y$ $\{y_1,y_2,\dots, y_n\}$. Lets say that entropy $H(X)=0$ i.e. $X$ has a probability distribution s.t. $P(X=x_j) = 1$ for only ...
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### Entropy of gamma-exponential compound distribution

Following this question, I have the PDF of a gamma-exponential compound distribution as $$f(y) = \frac{\alpha\beta^{\alpha}} {(y+\beta)^{(\alpha+1)}}$$ For my application I need the entropy of ...
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### Generalization of Shannon's source coding theorem with a posteriori entropies

This doubt is with reference to section 5-5 of "Information theory and Coding" by Prof. Norman Abramson. Under the topic "A generalization of Shannon's First Theorem", the text discusses how knowledge ...
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### mutual information and data processing inequality for $X\to Y\to Z$ where $Y=f(X)$

Let $X\to Y\to Z$ be three random variables. The data processing inequality states $I(X;Y)\geq I(X;Z)$. Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function. What ...
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### chain rule conditional entropy

I have to prove the chain-rule for conditional entropy. I kept getting stuck on one step, so I looked up a proof and found this: \begin{align}H(Y\mid X)&= \sum_{x\in\mathcal X, y\in\mathcal Y}p(x,...