I want to show that \begin{align*} (x^2\delta^{(3)},\phi) = -6(\delta^\prime,\phi) \end{align*} Where $\phi$ is a test function. I'm not quite sure how to go about it. My attempt was defining $f(x) = x^2\delta(x)$. I get that \begin{align*} (f^{(3)},\phi) = 6(\delta',\phi)+6(x\delta'',\phi)+(x^2\...

I have this Stochastic Process over here: $$ X_{t+1} = \begin{cases} X_t + \epsilon_t & \text{if } X_t > 0 \\ 0 & \text{if } X_t \leq 0 \end{cases} $$ $$ \epsilon_t \sim N(\mu_t, \sigma^2) $$ $$ \mu_t = -kX_t $$ I am trying to prove as to why when $k$ is between certain values, this stochastic p...

Given a fixed real number a>0: There exists a rational-into-rational function - f - such that: for all distinct and rational x,y the inequality | f(x) - f(y) | > a is satisfied And for all real-into-real functions - g -: there exists distinct and real x,y such that the inequality | g(x) - g(y) | <...

I am working through DeGroot Probability 4th Edition and came across two equivalent pdfs: The range of a sample of uniform random variables The second largest observation from a sample of uniform random variables I have restated the problem slightly, but the information and pdfs are below. Cons...

Let $X$ and $Y$ be independent, centered random variables. Let $\alpha \in (0,1)$ be fixed (e.g., $0.95$), and denote by $Q(X)$ the $\alpha$-quantile of $X$. I would like to determine under what conditions the following inequality holds: $$ Q^2(X+Y) \leq Q^2(X) + Q^2(Y). $$ I have identified one ...

At the bottom, there is an excerpt from John M. Lee's Introduction to Topological Manifolds. I am confused about: "Assuming this, it follows immediately that the geometric realization of $\langle S_1, a, b, c \mid W_1 c^{-1} b^{-1} a^{-1} \rangle$ is homeomorphic to $M_1 \setminus B_1$ (which we ...

Question. Let $n$ be an odd number. Do there exist unit vectors $v_1,\ldots, v_n$ in $\mathbb R^2$ such that both coordinates of each $v_i$ are rational and $$ v_1 + \cdots + v_n = 0 $$ If $n$ were even then it is easy give examples for the existence of the above. For $n=3$ I know that the ans...

Recently, I came across a very nice question from the 1985 AIME: Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at t...

When we integrate square root of a quadratic function, we can use method of completing the square along with suitable substitutions. Here I have listed some examples. $$\int{\frac{x-1}{\sqrt{x^2-2}}}dx$$ $$\int{\frac{1}{(x-1)\sqrt{x^2-2}}}dx$$ $$\int{\frac{\sqrt{x^2-1}}{x-2}}dx$$ For the first on...