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Let $X$ and $Y$ be two independent discrete random variables, and $Z$ be a function of $X$ and $Y$, i.e., $Z=f(X,Y)$. Suppose that $\Gamma$ is a set such that $$\mathrm{Pr}[(X,Y)\in\Gamma]\geq 1-\epsi …

Is there any closed form distribution formula for the distribution of the eigenvalues of $\mathbf{X}^\mathrm{H}\mathbf{X}$ where the entries of $\mathbf{X}$ are independent Gaussian random variables w …

Let $\mathbf{Z}$ be a $m\times n$ matrix with zero-mean unit-variance i.i.d complex Gaussian entries. What is the maximum value of mutual information $I(\mathbf{X}_1,\mathbf{X}_2;\mathbf{Y})$ such tha …

Let $\mathbf{U}$ be a random unitary matrix and $\mathbf{z}$ be a random i.i.d complex Gaussian vector (unitary invariant). Assume that the following relation is satisfied: \begin{align} \mathbf{y}=\m …

Let $\mathbf{H}_{N,K}$ be a random matrix whose entries are i.i.d complex Gaussian random variables with variance $1$. Then, we know from the law of large number that if $N,K\rightarrow\infty$, we hav …

Let $\mathbf{X}$ be a random matrix with independent Gaussian random variable entries with different variances $v_{ij}$. Also define $\mathbf{A}=\mathbf{X}^\mathrm{H}\mathbf{X}$. Is there any relation …

Assume that $\mathbf{X}$ is a random positive-definite matrix. Then, is there any upper or lower bound on the expectation of the following expression $$\mathbb{E}[\mathbf{X}^{-1}]-\alpha\mathbb{E}[\ma …

Let $X$ be a random variable which takes values in $\mathcal{X}$. Assume that we pass $X$ through two independent conditional pdf $p_{X_1|X}$ and $p_{X_2|X}$ and choose $X_1$ with probability $\lambda …

Let $X$ and $Y$ be two zero-mean independent and identically distributed random variables. Is there a bound for the following ratio, $$\frac{\mathbb{E}[|X+Y|]}{\mathbb{E}[|X|+|Y|]}=\frac{\mathbb{E}[ …

Let $X$ and $Y$ be two random variables with joint pmf $p(x,y)=p(x)\cdot p(y|x)$ and $X$ has uniform distribution. Also assume that the following relation is satisfied: \begin{align} \lVert p(y|x)-p(y …

Suppose $I(X;Y)$ denotes mutual information and on the other hand there is a relationship as follows. \begin{align} |p(y)-p(y|x)|<\delta p(y),\qquad\forall x,y. \end{align} Then we can say about mutua …

Suppose $X$, $Y$, $X'$ and $Y'$ are random variables whose probability density follows the following relations. \begin{align} \|p_X-p_{X'}\|_{\mathrm{TV}}&\leq\epsilon_1,\\ \|p_Y-p_{Y'}\|_{\mathrm{TV} …

Let $X$ be a random variable and $Y=f(X)$ where $f$ is a deterministic function. Moreover, assume that there exists a deterministic function $g(.)$ such that the following probability is small. \begin …

If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality: \begin{align} \mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\ …

Let $d_n, n\in\{1,2,\cdots,N\}$ be $N$ realizations drawn independent and identically from uniform distribution on $(0,L)$ where $L=\gamma\sqrt{N}$ with constant $\gamma$. Suppose that we need to appr …