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...

I am trying to evaluate the integral $$ f(x)=\int_0^{\infty}q^{2}\cos(2\pi qx)\,dq. $$ I first tried this using Mathematica, which said that the integral is divergent on $q\in (0,\infty)$. Looking at the problem again, the integrand grows quadratically and is not Riemann (nor Lebesgue) integrable...