All Questions
Tagged with pr.probability st.statistics
1,134 questions
1
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438
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Chain rule for maximal correlation
Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
- 11
1
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0
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65
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Normalizing constants preserve metric entropy
Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation
$$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
- 11
1
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1
answer
144
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Bounds for the extreme singular-values of random matrix with thresholded entries
Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
- 6,853
4
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1
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560
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Intuition behind the noncentral chi square as Poisson mixing
It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).
...
- 128k
1
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1
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370
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in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?
In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...
- 2,101
23
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7
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5k
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What makes Gaussian distributions special?
I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions.
...
- 850
0
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0
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86
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What probability distribution is this?
Thank you in advance for any suggestions or feedback.
I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$.
I am interested in finding the Wasserstein (...
1
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1
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276
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Upper bound for $P(X \geq x)$, where $X \sim \operatorname{Pois}(\lambda)$
I posted the following question in a comment on CDF of a log-concave discrete random variable. Since it is not related to my main question, I thought of reposting it as separate post.
Question:
Let $X ...
- 128k
2
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1
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118
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Calculate the discrete probability of x number of good outcomes occurring before y number of bad outcomes
I have a grid of 16 tiles face down. Half are good outcomes and half are bad outcomes. How would I calculate the probability of picking x number of Good outcomes before y number of bad outcomes are ...
- 842
3
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0
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98
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Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
- 279
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1
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266
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CDF of a log-concave discrete random variable
In the continuous setting, it's known that if a density function is log-concave , then its CDF is also log-concave.
My questions:
What can we say about this in the discrete setting?. For ex: Is the ...
- 128k
2
votes
1
answer
668
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Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k ...
- 6,853
1
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1
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84
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Lower bound on mean minimum distance($l_{\infty}$) between a test random vector $X'$ and vectors $X_1, \dots X_N$
Suppose we draw a independent random vector $X'$ uniformly from a unit hypercube, $[0, 1]^d$. Given similarly drawn vectors $X_1 \dots X_n$ we can define the following quantity
$\rho_{\infty}(d, n):= \...
- 128k
0
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2
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66
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Convergence of an orthormal expansion of the density
Suppose that $X_1,..,X_n$ are i.i.d real random variables with density $f \in L_2(\mathbb R)$, and that $g_i$ are function forming an orthonormal basis of $L_2(\mathbb R)$, i.e :
$$f(x) = \sum\limits_{...
- 128k
1
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0
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42
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Estimation of a density by orthogonal projection
I was wandering if the following problem was already treated in the litterature.
Suppose i want to estimate a density $f$ by an estimator $\hat{f}$ that i cannot explicitely describe.
All i have is ...
- 686
2
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3
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1k
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How can I prove Chebyshev's sum inequality with probabilistic methods?
I would like to prove Chebyshev's sum inequality, which states that:
If $a_1\geq a_2\geq \cdots \geq a_n$ and $b_1\geq b_2\geq \cdots \geq b_n$, then
$$
\frac{1}{n}\sum_{k=1}^n a_kb_k\geq \left(\frac{...
- 128k
7
votes
2
answers
524
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Projections in infinite dimensional statistical manifolds
I'm struggling to understand the geometry of projection for infinite dimensional statistical manifolds. In finite dimensions, a strictly convex smooth function $F$ defines a Bregman divergence. From ...
- 8,071
1
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0
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68
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(Anti-)concentration of gap between largest and second largest component of multivariate random gaussian vector
Let $n$ be a large positive integer and let $Y=(Y_1,\ldots,Y_n)$ be a zero-centered random $n$-dmensional real vector with covariance matrix $\Sigma$, an $n$-by-$n$ positive definite matrix with ...
- 6,853
8
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2
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5k
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Proof of Karlin-Rubin's theorem
I asked this question on Math Exchange, but as I did not receive a successful answer, maybe you could help me.
Karlin-Rubin's theorem states conditions under which we can find a uniformly most ...
CommunityBot
- 1
2
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2
answers
690
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Concentration and anti-concentration of gap between largest and second largest value in Gaussian iid sample
Let $n \ge 3$ be an integer and let $X=(X_1,\ldots,X_n)$ be random vector with iid coordinates from $N(0,1)$. For $1 \le k \le n$, let $X_{(k)}$ be the value of the $k$th largest coordinate of $X$.
...
- 128k
3
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1
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511
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Complete statistical manifolds
Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are some well-studied/interesting examples of statistical ...
- 6,001
1
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1
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517
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log-like distance between probability distributions
Given two probability density functions (PDF) $f$ and $g$, both defined over the same set $X$, there are many ways to describe/measure the distance between them, e.g., KL divergence and Hellinger ...
- 3,979
0
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1
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70
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Simulation of multivariate logistic distribution conditional to a plane
For an algorithm, I have to simulate $X_1, \ldots, X_n \sim_{\text{iid}} \text{Logistic}(0,1)$ conditionally to the event $(X_1, \ldots, X_n) \in P$, where $P$ is an affine plane in $\mathbb{R}^n$.
I ...
- 128k
1
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1
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261
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Concentration inequality for a function whose parameter depends on input samples
Concentration inequalities can be used to establish results such as sample mean cannot be too far from the actual population mean, and so on. For example, let $X_1 \ldots X_n$ be i.i.d instances of a ...
- 14.2k
1
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0
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49
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semi-parametric regression
Suppose the observation $(X_1, Y_1), \ldots, (X_n, Y_n)$ satisfies the following semi-parametric model
$$Y_t = m(X_t, \alpha) + \sigma(X_t, \beta) U_t,$$
where $U_t$ is independent with $X_t$ with ...
- 331
16
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5
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4k
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Is a fair lottery possible?
I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...
2
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2
answers
206
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non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 2,101
1
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0
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99
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Covering number after projection
In these lecture notes on Statistical Learning Theory we find the following definitions for covering numbers:
Definition. Let $(\mathcal{W}, d)$ be a metric space and $\mathcal{F} \subset \mathcal{W}$...
4
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1
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362
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Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
...
- 695
2
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0
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140
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Adding elements in a list *in expectation*
Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
- 43
1
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1
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519
views
The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
- 34.7k
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910
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Is Steven J. Miller's "research" on election fraud sound? And was he paid for it? [closed]
I recently encountered the following piece regarding alleged massive voter fraud in Pennsylvania:
https://justthenews.com/sites/default/files/2020-11/...
- 23k
-1
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1
answer
2k
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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 ...
- 28k
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1
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3k
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In linear regression, we have 0 training error if data dimension is high, but are there similar results for other supervised learning problems?
I tried posting this question on Cross Validated (the stack exchange for statistics) but didn't get an answer, so posting here:
Let's consider a supervised learning problem where $\{(x_1,y_1) \dots (...
- 128k
0
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0
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80
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Can we construct a surjective mapping from $\mathbb{R}^{?}$ to this space?
(Note : I'm not sure about the tags, please re-tag this if you think you have the right tag).
I am optimising a certain function over a certain space (that i will describe), and to use non-constraint ...
- 686
1
vote
1
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269
views
Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos?
I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form
$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$
where $\mu_{ijkl}$ are the ...
- 128k
12
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4
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2k
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Throwing a fair die until most recent roll is smaller than previous one
I roll a fair die with $n>1$ sides until the most recent roll is smaller than the previous one. Let $E_n$ be the expected number of rolls. Do we have $\lim_{n\to\infty} E_n < \infty$? If not, ...
- 3,937
1
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0
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35
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The asymptotic properties of $V$-statistic for mixing multivariate process
Suppose $\{X_t\}_{t \in \mathbb{Z}} \subseteq \mathbb{R}^d$ is a $\alpha$- or $\rho$-mixing process. Let $h (x, y) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the symmetric kernel ...
- 331
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1
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124
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Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, \infty).$
For each $ m \ge 1$, let $X_m$ and $Y_m$ be two non-negative iid random variables with the same distribution. (The distributions of $X_m$ may change with different $m$.)
**Assume that their support of ...
- 1,467
0
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1
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39
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The nonparametric estimation in generalized regression model
Let $Y_t \in \mathbb{R}$ be a response variable and $X_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X_1, Y_1), \cdots, (X_n, Y_n)$.
\begin{equation}
Y_{t} = \mu(...
- 331
2
votes
1
answer
649
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distribution on the inverse Wishart matrix eigenvalues summation
Let $\lambda_1>\lambda_2>....>\lambda_N$ be the ordered eigenvalues of Wishart matrix my objective is to find if it is possible the distribution of:
\begin{align}
s = \sum\limits_{i = 1}^...
2
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1
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1k
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Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$
Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
1
vote
0
answers
146
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Bounding the probability of success of adding elements into a list
Suppose $𝐿_1,…,𝐿_𝑘$ are lists with $n$ elements each. We use a fully independent hash function ℎ to compute a value for each element of each list. (We suppose the hash function returns a value ...
- 43
0
votes
1
answer
141
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Relaxing conditional independent assumption
Suppose we have random variables Y, D and X, where Y is independent of D conditional on X (Y⊥D|X). If there is another variable Z=f(X), where f(.) is a measurable real function, my question is: (1) ...
- 14.2k
2
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0
answers
49
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What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?
I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...
- 223
1
vote
1
answer
266
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Decomposition of the sum of nonnegative random variables [closed]
Non-necessarily independent random variables $X_1,~X_2,~\cdots,~X_n$ are supported on $[0,a_1],~[0,a_2],~\cdots,[0,a_n]$ and with mean values $\mu_1,~\cdots,~\mu_n$ respectively, where all $a_i$ and $\...
- 3,937
1
vote
1
answer
337
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Posterior expected value for squared Fourier coefficients of random Boolean function
Let $f : \{0, 1\}^{n} \rightarrow \{-1, 1\}$ be a Boolean function. Let the Fourier coefficients of this function be given by
$$ \hat f(z) = \frac{1}{2^{n}} \sum_{x \in \{0, 1\}^{n}} f(x)(-1)^{x \cdot ...
- 526
3
votes
1
answer
902
views
Expectation of exponential of Gaussian random matrix
Let $X$ be an $(N, M)$ random Gaussian matrix where $M<N$. For a given vector $v$, I want to estimate the expectation of:
\begin{align}
E\left[ {{v^T}X{X^T}{v}} \right]
\end{align}
and
\begin{align}...
- 188k
3
votes
2
answers
488
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Question about a new pseudo-random number generator
While investigating non-periodic RNG's (random number generators) for irrational numbers, I came up with a version that actually produces pseudo-random words consisting of $N$ bits, where $N$ is ...
- 3,098
0
votes
1
answer
496
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Laplace transform inversion
I have a probability distribution that is defined through it's Laplace transform by :
$$L(t) = \mathbb E(e^{-tX}) = e^{1 - \frac{1+t}{t}\ln(1+t)}$$
Using R and the invLT package, i have a numerical ...
- 128k