All Questions
Tagged with pr.probability probability-distributions
1,384 questions
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Laplace transform of a random variable: Inversion formula from an interval
Let $X$ be a non-negative random variable with a CDF $F$. Let $L_X(t)$ denote the Laplace transform of $F$, i.e.,
\begin{align}
L_X(t)=E[ e^{-tX}], \quad t \ge 0
\end{align}
It is known that $L_X(...
- 671
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1
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344
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How to prove that $X_1X_1', X_2X_2'$ are iid random matrices if we know that $X_1,X_2$ are iid random vectors? [closed]
Let $X_1, X_2$ be two iid random row vectors in $\mathbb{R}^p$, each of whose components are real valued. I'd like to prove that the $\mathbb{R}^{p\times p}$ random matrices $X_1X_1', X_2X_2'$ are ...
- 1,467
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2
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112
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Are these moments related to any usual distribution?
Knowing the moments of a random variable, we are wondering if we can express it in terms of some usual distribution (beta, uniform...), maybe as a product of distributions.
$X$ is a $[0,1]$-valued ...
- 11
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1
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222
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Does the inequality $\mu F(\mu)^2\geq \int_{\mu}^{b}F(x)\cdot(1-F(x))\,\mathrm dx$ hold for compactly supported continuous random variables?
This question was originally asked on the Mathematics StackExchange by User smcc
Consider a continuous random variable $V$ with cumulative distribution function $F$ and density function $f$. Suppose ...
- 1,326
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115
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Bayesian Bandits - What's the probability that choice K is the best?
I have $K$ very unfair coins. I don't know how unfair they are, but they all seem to have different probabilities of landing heads. I'd like to figure out which one is best as quickly as possible.
...
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Normality of the sum of uniformly distributed random variables
As noted in the recent answer by Yuval Peres, the sum of independent uniformly distributed random variables (r.v.'s) cannot have a normal distribution.
The question is, what happens without the ...
- 128k
2
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2
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801
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Weak convergence in Skorohod topology
Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability ...
- 411
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146
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Monotone coupling between "two-sided Gumbel" distributions
I am interested in finding a monotone coupling between two random variables. Let $\alpha_1>\alpha_2$, $b<a$. Define the following two (non-normalized) densities on the whole real line:
\begin{...
- 1,765
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85
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How fast does a sum of Bernoulli distributions (of different parameters) decrease after its mean?
Let $X=\sum_{i=1}^nX_i$, where each $X_i$ is a random variable following a Bernoulli distribution of parameter $p_i$. All $X_i$ are independent, and for all $i$, $p_i<p$ for some small $p$. I'm ...
- 267
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81
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Finding a distribution satisfying uncountably many constraints. Any relevant references?
The problem I'm dealing with has the following form. Let $X$ be some uncountable set, and $Y$ be some finite set. Suppose $f: X \times Y \to [0,1]$, and given $\mathcal{H} \subseteq Y^X$, I'm looking ...
- 33
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1
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285
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Exponential upper bounds for sums of martingale differences
Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...
- 65
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205
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Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
4
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272
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How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?
Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...
- 267
3
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2
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94
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An extremal value distribution from monotone sequences
Pick two integer sequences $d>a_n\geq a_{n-1}\geq\dots a_1\geq0$ and $d>b_n\geq b_{n-1}\geq\dots b_1\geq0$ where $d$ is an integer bound with following method:
Pick $a_n$ uniformly from $[0,d]$ ...
- 1,826
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344
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Degenerate Gaussian Integral
I have an integral over a subspace of $\mathbb{R}^n \times \mathbb{R}^n$ with an integrand of the form
$$\exp\left(-\frac{1}{2}\left[||u^2|| + \langle u, v \rangle + ||v||^2\right]\right)$$
The ...
- 435
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1
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66
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Comparing noisy truncated RV with noisy regular RV
For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
- 183
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429
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Trying to understand Fisher's proof
$\newcommand{\al}{\alpha}$
For $i=1,\dots,n$, let
\begin{equation}
R_i:=\frac{X_i}{X_1+\dots+X_n},
\end{equation}
where the $X_i$'s are iid standard exponential random variables. Let
$$R_*:=\max_{1\...
- 128k
1
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0
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81
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Empirical estimation of $\inf_{\gamma \in \Pi(\mu,\nu)}\gamma(\Omega)$, given i.i.d samples from $\mu$ and $\nu$
Let $\mathcal X$ be a Polish space and $\Omega \subseteq \mathcal X^2$ be open. Let $\mu$ and $\nu$ be probability measures, and consider the quantity $c_\Omega(\mu,\nu)$ defined by
$$
c_\Omega(\mu,\...
- 6,853
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1
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636
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Sufficient condition for function of conditional probability density to be increasing
Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
- 143
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1
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80
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What the probability of the max value of restricted random variable? [closed]
$x_1 + x_2 + \dots + x_n = 1, 0 \leq x_i \leq 1$, and $(x_1, x_2, \dots, x_n)$ evenly distributes on its restricted space, obviously which is a polygon on $n - 1$ dimension plane.
Let random variable ...
1
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1
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148
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algebraic tail of a random variable
Could anyone explain to me what does it mean by a map $f\to K_f$ and $f\to \rho(f(x_0), x_0)$ has an algebraic tail relative some measure, where $\rho$ is Prohorov metric, from this paper Paper which ...
- 149
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83
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Convergence in mean and convergence in distribution
Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that
$$
0< ...
- 411
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1
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1k
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Convergence in distribution of products
Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e.
$$
E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty.
$$
Moreover, there exist constants $c_0$ and $c_1$ such that
$$
0 &...
- 411
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60
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Concentration of Sample Mode
Is there a concentration bound for sample mode, when there exists a unique mode for a density $f$ that doesn't depend on $f$ (Essentially I am looking for a Chernoff type bound for mode)?.
- 53
2
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2
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538
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Probability space with exactly one Brownian motion
Very recently, the following question was asked:
Often, we encounder the assumption that $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ is a stochastic base on which a Brownian motion is defined. ...
- 128k
1
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1
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88
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Samples from a modified Bernoulli
Given i.i.d samples $X_1, X_2, \cdots$ from Bernoulli($p$) and $1<c<\frac{1}{p}$, is it possible to construct samples from Bernoulli($cp$) under the assumption that $p$ is unknown?
If $c\leq1$ ...
- 53
1
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1
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155
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Reference request concerning order statistics from the uniform distribution
Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\...
- 128k
4
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1
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812
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On the largest and smallest spacings for the uniform distribution
Let $Z_1,\dots,Z_n$ be iid random variables (r.v.'s) each uniformly distributed on $[0,1]$. Let $Z_{n:1}\le\cdots\le Z_{n:n}$ be the corresponding order statistics. For $i=1,\dots,n-1$, let $G_i:=Z_{n:...
- 128k
222
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Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?
Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
- 2,272
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1
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160
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Central limit type theorems for compact Hausdorff topological groups?
Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure
$(\tau*\mu)(A)=\int\int1_A(xy)d\tau(x)...
- 646
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316
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Wasserstein distance between rotated conditional distributions
Suppose we have a probability distribution $\rho$ on $\mathbb{R}^d$. Let $ E \subset \operatorname{supp}(\rho) $, and $R_\theta$ a rotation of angle $\theta$ such that $ R_\theta E \subset \...
- 11
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157
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Marginal probability mass function
I have the joint PMF
$\exp(y_1 \ln(\lambda)+y_2 \ln(c)+y_2\ln(\lambda)-\ln(y_1!y_2!)-\lambda(1+c))$
for a constant $c>0$. In canonical representation and mixed parameterization I have $\mathbf{\...
- 111
1
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1
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273
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Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)?
Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x_i$ generated by RAND() function of Excel). I ...
- 51
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1
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"Сross сubic variation" of two Brownian motions and interpretation of the simulation result
Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$.
How to calculate the expression below? Can we rewrite ...
- 21
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Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings
Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means.
Question. Given $\alpha > 0$, what is value of, ...
- 6,853
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Can we show that $\mathbb R^{\mathbb N}\ni x\mapsto\bigotimes_{n\in\mathbb N}\mathcal N_{x,\:\sigma^2}$ is a Markov kernel?
Let $\sigma>0$ and $\mathcal N_{x,\:\sigma^2}$ denote the normal distribution with mean $x\in\mathbb R$ and variance $\sigma^2$. From the Ionescu-Tulcea theorem, we know that $$\kappa(x,\;\cdot\;):=...
- 167
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Uniqueness of martingale problem for Levy type operator
Consider the following Levy type operator:
$$
L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d),
$$
...
- 411
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208
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On the difference of conditional differential entropy of two correlated random variables
Problem Definition
Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where
$\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...
- 31
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364
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Can anyone give a reference to the proof of this concentration inequality?
The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%...
- 123
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Existence of stationary stochastic processes with very high correlation
A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. ...
- 128k
3
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Estimating the probability density of a component of a mixture distribution
Let $X \in \mathbb{R}^d$ be a random variable with probability distribution $P$. Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be an invertible function and let $P_{f}$ be the distribution of random variable $...
- 31
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3
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536
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On exponential distributions and dot products
Let
$a, b$ be two variables drawn from an exponential distribution with parameter $\lambda_1$.
$c, d$ be two variables drawn from an exponential distribution with parameter $\lambda_2$.
I am ...
3
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0
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177
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Does this property of probability distributions have a name?
I'm working on the paper where we require for our distributions to be somewhat dense even at very small left-tail values.
The idea is that the CDF of our distribution should grow, from the very ...
- 396
2
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1
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185
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Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$
Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.
...
- 6,853
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1
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74
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Joint density of a quadratic function of entries of orthogonal matrix
$U=(U_{ij})_{1\leq i,j\leq m},V=(V_{ij})_{1\leq i,j\leq m}$ are independently and uniformly distributed on the orthogonal group $O(m)$. For any positive integer $k,n$ such that $1\leq k\leq n\leq m$, ...
- 1,495
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0
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244
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An inequality regarding centered Bernoulli random variables
Let $\left\{V_1,\ldots,V_n\right\}$ be a set of (possibly dependent) identically distributed Bernoulli random variables, with
$$ p = \mathbb{P}\left(V_i=1\right) = 1-\mathbb{P}\left(V_i=0\right),\quad ...
- 159
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1
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248
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Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
- 1,495
14
votes
1
answer
813
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Large-n limit of the distribution of the normalized sum of Cauchy random variables
What is the large-n limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? ...
- 419
4
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0
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212
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A lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution
Given a set of Bernoulli random variables $x_1, \dots, x_n$ (not necessarily identical) with $X= \sum_{0<i\leq n} x_i$, I am intrested in finding a lower-bound for $\frac{\mathbb{E} [ \min (X,k) ]...
- 189
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0
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112
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Hashed coupon collector
The story:
A sport card store manager has $r$ customers, that together wish to assemble a $n$-cards collection.
Every day, a random customer arrives and buys his favorite card (that is, each customer ...
- 11