All Questions
18,177 questions
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votes
1
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332
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Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
- 1
-3
votes
3
answers
628
views
Roulette probability [closed]
I'm looking for some knowledge on probability, I've scoured the net but I can't really grasp the answer.
I was having a discussion with a co-worker about roulette probability. He says that at any ...
-3
votes
1
answer
144
views
Count arrangements with pairs of attacking kings [closed]
I have a $1\times n$ chessboard and $2$ pairs of kings in it. Both components of each pair of kings must be adjacent in the chessboard, that is, they must be attacking.
Now, I want to calculate the ...
- 47
-3
votes
1
answer
123
views
Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
- 11
-3
votes
2
answers
156
views
Getting almost certainty from uncountably many low-probability events
Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
- 5,405
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votes
2
answers
450
views
Expected values of two random variables related to a simple urn problem
In an urn there are $u$ balls, $b$ of which are black.
If we perform $n$ trials of one ball at a time with replacement, the probability of the event $E$ to get $n$ times a black ball is $P(E)=\left(\...
- 145
-3
votes
1
answer
392
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
-3
votes
1
answer
960
views
how to formalize a notion of symmetric set difference probability? [closed]
I saw in a paper an argument, that seems simple. (Let $\triangle$ be the triangle operator of symmetric set difference between two sets)
It states that if $P(A \triangle B) < \epsilon$, for some ...
-3
votes
1
answer
440
views
Conditional expectation [closed]
Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
- 19
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
-3
votes
1
answer
117
views
Combinatorial meaning of a reduced fraction in a simple probability problem?
A routine exercise for undergraduates says: Given that the number of successes in $20$ independent Bernoulli trials was $8,$ what is the conditional probability that exactly $3$ of those $8$ successes ...
-3
votes
1
answer
63
views
How to show $\lambda_i \in \sigma_A(x)$?
Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
- 33
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votes
0
answers
11
views
Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
- 1
-3
votes
0
answers
66
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-3
votes
0
answers
136
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619
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votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
-3
votes
1
answer
318
views
Porbability of selecting balls from boxes [closed]
There are three boxes. B1, B2, B3 The probability of selecting them is 0.2, 0.2 , 0.6 respectively.
B1 contains 3 red balls and 7 green balls.
B2 contains 5 red balls and 5 green balls.
B3 contains ...
-4
votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,064
-4
votes
2
answers
286
views
Does the Laplacian commutes with the indicator function [closed]
We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
-4
votes
1
answer
370
views
Is delta function symmetric against real axis? [closed]
Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?
I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis.
We can write Delta function as
$$\delta(z) = \...
- 10.1k
-4
votes
1
answer
66
views
Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$? [closed]
Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that
$$
\sup_n\int_{E}|f_n|d\mu<+\infty.
$$
Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
- 115
-4
votes
1
answer
144
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
-5
votes
2
answers
648
views
Another question on Øksendal's book
Hi
On page 98 "Stochastic differential equations" of Øksendal, 6th edition,
the author writes that $$\int_{0}^{u}\Big(\int_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR_{s}\Big)dt=\int_{0}^{u}\Big(\...
- 281
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votes
1
answer
149
views
Lottery in O(1) per participant
Goal: implement in $O(1)$ per participant a lottery where each participant has some large number of tickets, and the best (e.g. least) one wins, without actually burning electricity in proportion to ...
- 99
-6
votes
2
answers
2k
views
Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307