## Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique

The density function of the distribution function of a continuous random variable is not uniquely defined.
A new density function can be obtained by changing the value of the function at finite number of points to some non-negative value, without changing the integral of the function. We then get a new density function for the same continuous distribution.

Does this follow from the theorem-

A bounded function with finite number of discontinuities over an interval is Riemann integrable.

or is there a different theorem supporting the above claim? Is the theorem a sufficient justification?

## Density of numbers with multiple factors near square root

Fix constants $$1\leq \alpha<\beta$$. What is the density of the set of positive integers $$n$$ with at least two factors between $$\alpha\sqrt{n}$$ and $$\beta\sqrt{n}$$?

(I am specifically interested when $$\alpha=\sqrt{2}$$ and $$\beta=\sqrt{3}$$, and I am hoping the density is zero. I am not an expert in this field, so apologies in advance.)

## probability density function convolution theorem

I have trouble with solving this: Let the joint density of X and Y be f(x,y)= 6e^(-3x-2y) if x>0 and y>0, and 0 otherwise.

Let Z = X + Y. Use the convolution theorem to find a density for Z. thank you

## Spectral density of $D + XX^T$

Let $$D$$ be a fixed diagonal matrix with real entries, and $$X$$ a random $$m\times n$$ matrix. More precisely, the entries $$X_{ij}$$ are real independent and identically distributed. It can be shown that the eigenvalues of $$XX^T$$ have a density following the Marcenko-Pastur law with probability 1, in the limit that $$n,m\rightarrow\infty$$ with a fixed ratio $$m/n$$, and this result depends only on the variance of the entries $$X_{ij}$$.

Can we say something about the spectral density of $$D + XX^T$$, also in the limit of large $$m,n$$? Is an analytical expression (as in the Marcenko-Pastur law) known in this case? At least something numerically tractable?

## Solving density matrix equations

I am trying to evaluate Numerical Integration of a density matrix element by solving coupled density matrix equations for different values of certain parameter L ( find the code in the link). My code is working fine for L=0, but not L =1,2,…

## Definition of model functions and their density in $C^0(X^\text{an})$

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/656, arXiv:1201.0187).

Here is another question i have: In subsection 2.5 they define a model function as a continous $$\varphi$$ on $$X^{\text{an}}$$ s.t. there is a vertical Cartier Divisor $$D \in \text{Div}_{\mathbb{Q}}(\mathcal{X})$$, where $$\mathcal{X}$$ is a model of $$X^{\text{an}}$$, with $$\varphi_D = \log \vert \mathcal{O}_{\mathcal{X}}(D) \vert = \varphi$$. The set of these functions is denoted by $$\mathcal{D}(X) = \mathcal{D}(X) _{\mathbb{Q}}$$. In Prop. 2.2 they then show, among other things, that $$\mathcal{D}(X) _{\mathbb{Z}}$$ is stable under max. and seperates points. (I am assuming in the definition of $$\mathcal{D}(X) _{\mathbb{Z}}$$ one just considers $$D \in \text{Div}_{\mathbb{Z}}(\mathcal{X})$$?)

Here are my Questions: Is it clear, that if $$\mathcal{D}(X) _{\mathbb{Z}}$$ seperates Points and is stable under max. (see Prop.2.2), that this is also true for $$\mathcal{D}(X) _{\mathbb{Q}}$$?

Is there an isomorphism from $$\mathcal{D}(X) _{\mathbb{Q}}$$ to $$\mathcal{D}(X) _{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes}\mathbb{Q}$$?

Is it immediate that the $$\mathbb{Q}$$-VS $$\mathcal{D}(X) _{\mathbb{Q}}$$ fulfills the conditions of the Stone-Weierstraß Theorem(see Cor.2.3) and is thus dense in $$C^0(X^{\text{an}})$$? From my understanding one would require that $$\mathbb{Q}$$-VS $$\mathcal{D}(X) _{\mathbb{Q}}$$ is closed uner multiplication with elements from $$\mathbb{R}$$?

Thanks a lot guys.

## Eigenvalue density of $S^TS + ee^T$

Let $$S$$ be a random $$M\times N$$ matrix with independently identically distributed entries. The Pastur-Marcenko law gives the spectral density of $$S^T S$$ as $$N\rightarrow\infty$$ with a fixed ratio $$\alpha=M/N$$:

$$\rho_{\textrm{MP}} (\lambda) = \left\{\begin{array}{ll} \frac{\alpha}{2 \pi \lambda \sigma^2} \sqrt{(b – \lambda) (\lambda – a)} + (1 – \alpha) \delta (\lambda) & a < \lambda < b\ 0 & \textrm{otherwise} \end{array}\right.$$

where $$a,b=\left( 1 \pm \sqrt{\alpha} \right)^2 \sigma^2 / \alpha$$.

Let $$e_1$$ be the Cartesian unit vector in a fixed direction and consider the rank-one updated matrix $$A = S^T S + e_1e_1^T$$. The spectral density of $$A$$ has an analytical formula in the large $$N$$ limit?

## Probability density from standard domain (Typical Box principle and Chinese Remainder Theorem) – III

Pick a random pair $$(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$$. Denote $$N_2(a,b,n)$$ to be minimum $$\ell_r$$ norm of vector $$(x,y)$$ as $$(x,y)$$ ranges over all non-zero integral solutions to $$(x,y)\equiv t(a,b)\bmod n$$ where $$t\in\mathbb Z$$ with $$0.

Now let integers $$a,b$$ be of size in $$(\sqrt m,m-\sqrt m)$$.

Let $$p_1,p_2$$ be primes of similar size some integer $$m$$ and let $$t_1$$ attain $$N_2(a,b,p_1)$$ and $$t_2$$ attain $$N_2(a,b,p_2)$$. From Dirichlet pigeonhole we know $$N_2(a,b,p_1)$$ and $$N_2(a,b,p_2)$$ are less than $$\sqrt{2m}$$.

Let $$t$$ attain $$N_2(a,b,p_1p_2)$$ (note $$(t,p_1p_2)=1$$ need not hold while $$(t_1,p_1)=(t_2,p_2)=1$$ holds).

Then $$N_2(a,b,p_1p_2)$$ seems to be typically $$m\sqrt 2$$.

1. Is it possible for $$N_2(a,b,p_1p_2)$$ to be smaller than $$\sqrt{2m}-\mu$$ if $$N_2(a,b,p_1)$$ and $$N_2(a,b,p_2)$$ are greater than $$\sqrt{2m}-\mu$$ at a $$\mu\in(0,\sqrt{2m})$$?

2. If so what is the probability?

This is the result from Akshay Venkatesh when $$n$$ is prime. Then it is true as $$n\rightarrow\infty$$ the distribution of $$N_2(a,b,n)/\sqrt{n}$$ coincides with distribution of $$1/\sqrt y$$ where $$x+iy$$ is picked at random with respect to hyperbolic measure from $$\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$$.

## Given two probability density functions find a number that satisfies a given equation

I have a problem for which I either need a proof or a counterexample.

We are given two discrete random variables $$x_1$$ and $$x_2$$ in $$[0, n]$$ where $$F_1(x)$$ is the probability of $$x_1\leq x$$, and similarly $$F_2(x)$$ is the probability of $$x_2 \leq x$$. My goal is to show that there always exists a number $$c > 0$$ that satisfies $$\int_{0}^{\infty}x{f_2(x)}(1-F_1(x))\leq c\int_{c}^{\infty}f_1(x)F_2(x),$$ where $$f_1(x)= \Pr[x_1=x]$$ and $$f_2(x)=\Pr[x_2=x]$$.

Thanks for any help.