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Consider continous random variable $X$ with bounded support $\mathcal{X} \subset \mathbb{R}$, discrete random variable $Y$ with finite support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(x,y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ and $X_2$ are i.i.d. realizations from the probbaility measure on $X$. What are $c_1,c_2$ that maximize $I(Y;Z)$ where $I$ is the mutual information? what about general case with $n$ constants $c_1,...,c_n$?

Consider continuous random variable $X$ with bounded support $\mathcal{X} \subset \mathbb{R}$, discrete random variable $Y$ with finite support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(x,y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ and $X_2$ are i.i.d. realizations from the probability measure on $X$. What are $c_1,c_2$ that maximize $I(Y;Z)$ where $I$ is the mutual information? what about general case with $n$ constants $c_1,...,c_n$?

Consider discrete random variable $X$ with finite support $\mathcal{X}$, continuous random variable $Y$ with bounded support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(x,y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ and $X_2$ are i.i.d. realizations from the probbaility measure on $X$. What are $c_1,c_2$ that maximize $H(Y,Z)-H(Y\mid Z)$? what about general case with $n$ constants $c_1,...,c_n$?

Consider continous random variable $X$ with bounded support $\mathcal{X} \subset \mathbb{R}$, discrete random variable $Y$ with finite support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(x,y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ and $X_2$ are i.i.d. realizations from the probbaility measure on $X$. What are $c_1,c_2$ that maximize $I(Y;Z)$ where $I$ is the mutual information? what about general case with $n$ constants $c_1,...,c_n$?

Consider discrete random variable $X$ with finite support $\mathcal{X}$, continuous random variable $Y$ with bounded support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(X,Y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ nd $X_2$ are i.i.d. realizations from the probbaility measure on $X$. What are $c_1,c_2$ that maximize $H(Y,Z)-H(Y\mid Z)$?

Consider discrete random variable $X$ with finite support $\mathcal{X}$, continuous random variable $Y$ with bounded support $\mathcal{Y}$ , and constants $c_1 ,c_2 \in \mathbb{R}$. Let $p_{XY}(x,y)$ be the probability measure on $\mathcal{X} \times \mathcal{Y}$. Define $H(X,Y)$ and $H(Y\mid X)$ as the joint and conditional entropies. Define $Z = c_1 X_1 + c_2 X_2$ where $X_1$ and $X_2$ are i.i.d. realizations from the probbaility measure on $X$. What are $c_1,c_2$ that maximize $H(Y,Z)-H(Y\mid Z)$? what about general case with $n$ constants $c_1,...,c_n$?