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I’m trying to better understand the continuous spectrum of $ G = \operatorname{GL}_2(\mathbb A_{\mathbb Q})$ , which is the direct integral of induced representations $ \mathbf H(s) = \operatorname{Ind}_{Z(\mathbb A)N(\mathbb A)T(\mathbb R)^0T(\mathbb Q)}^{G(\mathbb A)} \omega_s$ . The reference I’m using is Forms of GL(2) from the analytic point of view by Gelbart and Jacquet inRead more

Observe the following code: p = 0.5; n = 10^2; XbarObservationSet = Mean[RandomVariate[GeometricDistribution[p], {n, n}]] myplot1 = ????? myplot2 = Plot[PDF[NormalDistribution[0, 1], x], {x, -3, 3}] Show[myplot1, myplot2] Suppose my goal is to take all of the data collected in XbarObservationSet and display an approximated, continuous probability density function corresponding to the observed data andRead more

Suppose I define a function in mathematica that has all of the properties of a valid probability density function. Would it be possible to have Mathematica simulate a random sample of a given size of values from this PDF? For example, if I define a function which is the PDF of a beta distribution, canRead more

I want to plot an arbitrary 2D function, for example $ e ^ { -(x^2+y^2) } $ . The plot should contain dots, the density of dots at a point indicating the value of the function at that point. The output should be something like this: How to do it?Read more

Suppose X is a random variable that we can observe, we want to take an action Y in order to minimize $ $ E[(Y-X)^{2}]$ $ Given the density of X is $ p(x)$ , what is the optimal choice of the conditional density $ q(y|x)$ of $ Y|X$ . Formally: $ $ \min_{q}\{E[(Y-X)^{2}]=\int(y-x)^{2}q(y|x)p(x)dydx\}$ $ s.tRead more

I am wondering whether the following result is true: Let $ \mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $ p$ and let $ \eta$ and $ \gamma$ be two probability measures in $ \mathcal W_p(\mathbb R^d)$ , such that supp($ \eta$ ) $ \subset$ supp($ \gamma$ ). Then there is a sequence ofRead more

To simplify the question, we start with standard Brownian motion(BM) $ B_t$ . Then, $ $ \lim_{\epsilon \to 0} \frac{1}{\epsilon}\mathbb P( 1- \epsilon <B_1 < 1 ) = \phi(1),$ $ where $ \phi$ is the density function of the standard normal distribution. Next, let $ B^1_t$ be the killed BM at $ 1$ , i.e.Read more

What is keyword density?Read more

what is keyword density?Read more

I have 2d points and want to show a colored density plot of them. Also I want to display a legend showing which color corespons to which total number of points in a certain predefined area. Let’s take the following 2d points: SeedRandom[1]; points = Transpose[{RandomReal[2, 100], RandomReal[3, 100]}]; ListPlot[points, Frame -> True, PlotRange ->Read more