All Questions
9,497 questions
-1
votes
1
answer
199
views
Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
- 167
-1
votes
1
answer
259
views
Which type of convergence for this sequence of random variables? [closed]
Suppose that $X_1,X_2...$ is a sequence of non-negative real random variables. I have that $\mathbb{E}(X_i^2) \to 0$ as $i \to +\infty$, therefore my sequence converges at least in distribution to the ...
- 686
-1
votes
1
answer
163
views
Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?
I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors.
Exercise 3.1.7 from the book is the following.
Let $X =...
- 401
-1
votes
1
answer
222
views
Why do we define independence for zero-probability events? [closed]
I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of ...
- 9
-1
votes
1
answer
175
views
A strange probability inequality
I need help to understand the following :
For any non-negative random variable $\zeta$: $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)...
- 1
-1
votes
1
answer
88
views
Can we say that: $ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $
Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ two $L^1$-bounded sequences, such that :
$$
\sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\...
-1
votes
1
answer
331
views
Expectation where linearity does not hold
We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude ...
-1
votes
1
answer
338
views
about Function of Random variables [closed]
Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤...
- 117
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
-1
votes
1
answer
148
views
Weighted sum of zero-mean random variables
Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
- 367
-1
votes
1
answer
304
views
The category Prob of finite measure spaces does not admit all products [closed]
I am currently working in a category called Prob which has objects which are finite measure spaces and morphisms which are measure preserving maps between the spaces. A map $f:X\to Y$ is measure ...
- 91
-1
votes
1
answer
172
views
The distribution of the sum of a non-zero vector with random signs
Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...
- 121
-1
votes
1
answer
129
views
Is it possible to regress an arbitrary function from a span?
Suppose we have three sets $A, B,C$ and a span $S := A \leftarrow C \rightarrow B$. There is a special case when the data of the span $S$ exactly specifies a function $f: A \rightarrow B$. In ...
- 1,313
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
-1
votes
2
answers
614
views
Bounded difference functions and sub-Gaussian random variables
We have the following standard theorem : Let $X$ be some set and $g : X^n \rightarrow \mathbb{R}$ be a measurable function such that it satisfies the ``bounded difference property" i.e $\exists$ $\{...
- 2,246
-1
votes
1
answer
197
views
Surely recurrent random walks and the law of the iterated logarithm [closed]
Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with
$$
P(X_i=1)=P(X_i=-1)=1/2,
$$
and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As ...
- 965
-1
votes
1
answer
558
views
For i.i.d X and Y , if X + Y and X - Y are independent, show X is normally distributed [closed]
The question goes as follows:
If $X$ and $Y$ are independent and identically distributed, their density function $f(x)$ is strictly positive and second-order continuously differentiable. If $X+Y$ and $...
- 9
-1
votes
2
answers
462
views
How to deal with this Chicken-And-Egg problem ?
Let's imagine designing an odds pattern for a game, in which players bet for win or lose.
Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.
Now imagine $n_1$ people ...
- 99
-1
votes
1
answer
1k
views
Approximating expectation [closed]
if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points:
E[f]=(1/N)(summation of f(x) over these N points).
...
-1
votes
1
answer
247
views
Concentration results for non-standard Gaussian random vectors.
Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) ...
- 11
-1
votes
1
answer
571
views
Formal definition of 'useful' ?
Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
- 11.8k
-1
votes
1
answer
80
views
Seating assignment inspired question
Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course ...
- 48.8k
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
-1
votes
1
answer
503
views
Minimum of exponential distribution
Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...
- 9
-1
votes
1
answer
128
views
Large deviation of sum of Gaussian variables
Let $X_i$ is $N(b_i,b_i^2)$, where $0<b_i\leq \alpha$. The $X_i$ are independent, but not identical (i.e. $b_i$ are not all equal). We concern the upper bound of the tail probability $P(|\sum_{i=1}^...
- 405
-1
votes
1
answer
98
views
Distribution of root with Poisson leaves
We have a pool of items, termed as item A, generated following a Poisson distribution. We use a pair of items A to produce an item B with success rate $r\in(0,1)$. My question is: is B Poisson ...
- 367
-1
votes
2
answers
157
views
Cumulants of a sequence of variables with zero mean and variance
Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?
- 13
-1
votes
1
answer
90
views
A periodically independent stochastic process
Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X_t$ independent of $X_{t+1}$ for all $t \geq 0$?
- 6,155
-1
votes
1
answer
2k
views
how to prove that the real part and the modulus of a characteristic function is still a characterisitc function? [closed]
this is a problem from Durret's probability textbook.
Show that if $\varphi$ is a ch.f., then $Re\varphi$ and $|\varphi|^2$ are also ch.f.
I am wondering how to prove this. Actually I'm not even sure ...
- 11
-1
votes
2
answers
1k
views
Does the variance of a strictly monotonically increasing function of a random variable have anything to do with the variance of the random variable? [closed]
Assume that there is a continuous random variable x, and its variance is var(x). Furthermore, there is a strictly monotonically increasing function f. Can anybody prove that the larger the var(x), the ...
- 1
-1
votes
2
answers
114
views
On OR condition in Linear Programming with exponentially many constraints [closed]
Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
- 13.9k
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 19
-1
votes
1
answer
429
views
What is the expected area of the triangle? [closed]
We create the unit equilateral triangle and put one vertex on each side the of the equilateral triangle and then connect them. What is the expected value of the triangle formed by the connection of ...
-1
votes
1
answer
95
views
Proving maximal entropy [closed]
It is quite easy to prove that
$$H(S) \leq \log_2(|A|),$$
where $A$ is the number of events, using the Jensen inequality
$$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) =...
-1
votes
1
answer
186
views
Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? [closed]
This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X_i$ with different occurrence probabilities, $\mathrm{Pr}[X_i=1]=p_i>p_{\min}>0$ ...
- 307
-1
votes
2
answers
438
views
Are the coefficients of a linear combination of random vectors as random?
Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
- 271
-1
votes
1
answer
213
views
Regarding a new divergence function of two probability distributions
Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any $0 \leq \alpha \leq 1$, and any constant $\beta$ within the support of $X$ and $Y$ such ...
- 482
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
-1
votes
1
answer
287
views
Property of relative entropy [closed]
For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...
- 35
-1
votes
1
answer
545
views
probability mass function fitting [closed]
I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)
![alt text][1]
[image shack image removed]
(...
- 307
-1
votes
1
answer
502
views
Name for probabilistic version of Pascal's identity and differentiation formula for binomial distribution
I'm trying to find a standard name or standard reference for two simple-to-prove relations involving binomial distributions.
Define:
$b(n,r,p) := \binom{n}{r}p^r(1 - p)^{n-r}$
i.e., it is the ...
- 7,320
-1
votes
1
answer
92
views
Variance of bins for N balls into M bins [closed]
If I throw N balls independently into M bins with uniform probability, the expected mean of the M bins is N/M balls.
What is the expected variance of the M bins?
I was thinking of what bin size I ...
-1
votes
1
answer
103
views
Convergence in $\mathbb{L}_1$ implies convergence "perturbed" conditional expectations
Consider a sequence of conditional pdf's $p_n(y | x)$ on a Polish space $X \times Y$, endowed with its Borel sigma algebra. Suppose, as $n\rightarrow \infty$, in $\mathbb{L}_1$ (the following ...
- 141
-1
votes
1
answer
148
views
Strong law of large numbers for a sequence of random variables in different probability spaces
Is it known whether the following version of the strong law of large numbers holds?
For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...
- 1
-1
votes
1
answer
169
views
joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
-1
votes
1
answer
273
views
What does $\mu$ and $\nu$ "dependent" mean? [closed]
On the other hand, if $\mu$ and $\nu$ are completely dependent then $\pi_{x_1}=\delta_{f(x_1)}$ for some function $f$. Then $W(\pi)=1$.
Note that
$$
\pi(dx, dy)=\pi_x(dy)\mu(dx).
$$
If $\pi_x(dy)=\...
- 288
-1
votes
1
answer
159
views
Poisson distribution and conditional expected value [closed]
I have a task:
Lat's take independent variables $X_{i}$ with Poisson distribution $Poiss(a)$. Distribution of $a$ has density $p(a)=\frac{8}{3}a^3e^{-2a}, a\ge0$.
Calculate:
$E(a|X_1=3, X_2=2, X_3=5, ...
- 101
-1
votes
1
answer
95
views
Multi-variable expansion of $e^{u^T X}$ as a power series in terms of tensors [closed]
I have the following question related to multivariable moment generating functions. Given two vectors $u,X\in\mathbb{R}^d$, I want to characterize the quantity $e^{u^T X}$ in terms of "powers&...
- 9
-1
votes
1
answer
92
views
Is the distribution of a Banach space valued Lévy process uniquely determined by its characteristic function?
Let $E$ be a $\mathbb R$-Banach space. Remember that if $\mu$ is a finite measure on $\mathcal B(E)$ then $$\Phi_\mu:E'\to\mathbb C\;,\;\;\;\varphi\mapsto\int\mu({\rm d}x)e^{{\rm i}\varphi(x)}$$ is ...
- 167
-1
votes
1
answer
164
views
Covariance inequality with Lipschitz functions
Suppose that $X$ and $Y$ are random variables and suppose that for all Lipschitz functions $f$ and $g$ s.t. $f(X),g(Y)\in L^p$, $p>2$,
$$
|\operatorname{Cov}(f(X),g(Y))|\le \big(\operatorname{Lip}(...
- 99