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I am struggling with the following problem. Let $f$ be a real smooth function: strictly convex on $(-\infty,0)$, strictly concave on $(0,\infty)$, strictly increasing. For $\sigma>0$, how can one ...

Let $f$ be a : $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$, for all $x> 0,~f(x)>0$, for all $x< 0,~f(x)<0$, I am struggling to find a bound for the distance between the root of $f$ ...

$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...

Given a convolution integral $$ g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...

There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector. Now I want to blur (convolve) it along with $u$ by another 2D ...