Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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What's the cumulative probability of these particular bags of liquorice allsorts?
After eating a bag of liquorice allsorts in one sitting, as one does, I noticed that it had contained an unusual amount of brown ones (which, you will agree, are an abomination that should never have ...
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When can a convolution be written as a change of variables?
Suppose $X$ is a random variable with a density $f(x)$ such that $f(x)$ is a convolution of some density $g$ with some other density $q$:
$$
f = g\ast q.
$$
Under what conditions does $X=h(Y)$, where $...
- 5
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How to compute the unique disintegration w.r.t. the first coordinate?
Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
- 288
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112
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A quadratic optimization problem involving Brownian motion
I wonder how to solve the following quadratic optimization problem?
$$\min_{f:[0,1]\to \mathbb{R}} \int_0^1 f(t)^2dt-2\int_0^1 f(t)dB_t$$
where $B(t)$ is standard Brownian motion.
Intuitively, the ...
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247
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Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,677
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2
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207
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Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality?
I have a question on Lawler – Notes on the Bessel process, on page 4. Let $X_t$ be one-dimensional Brownian motion, and we want to use $N_t$ as a measure-changing (local) martingale, defined as $$N_t=\...
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376
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Random variable is Big O in probability notation
Let $R_n$ be a random variable with values in $[0,1]$ and $nR_n$ converges to $\frac{1}{1+C} \chi_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-...
- 115
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2
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187
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Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$
Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...
- 6,853
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133
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Expectation value of random walk on $\mathbb{R}$ [closed]
Given iid random variables $\xi_1,\dots,\xi_n,\dots$ with probability $P(\xi_1=-\log 3)=\frac{1}{2}=P(\xi_1=\log 5-\log 3)$. Let $S_n=\sum_{i=1}^n \xi_i$. This is a random walk on $R$. If let a ...
- 288
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151
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Construction of a probability measure from a sequence of probability measures
Summary
I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.
Details
We work on ...
user475819
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217
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Is a sum of a bounded random variables the same order as its standard deviation?
The Question
Suppose that $X_n$ are independent random variables, $|X_n| \leq 1$, and $\mathbb{E}[X_n] = 0$. Let $S_n = \sum_{i=1}^n X_i$ and let $s_n = \sqrt{\sum_k \sigma^2(X_k)}$ be the standard ...
- 143
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133
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How to demonstrate a correlation inequality? [closed]
If there are 3 vectors X, Y, Z of the same length, for any $x_i \in X,y_i \in Y,z_i \in Z$, we have $0<x_i<1,0<y_i<1,0<z_i<1$.
The correlation between Z, Y is greater than between X, ...
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83
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The distribution of number of reverse order pairs in a randomly permuted array
There is an array $a_1,\dotsc,a_n$ whose elements are pairwise distinct. We define a reverse order pair to be an ordered pair $(a_i,a_j)$ such that $i < j$ and $a_i > a_j$. Consider the total ...
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175
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CLT for bounded difference functions
Let $X_1, \ldots, X_n$ be independent and identically distributed random variables. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a bounded difference function, i.e., for any $x,y \in \mathbb{R}^n$ that ...
- 519
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2
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204
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What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?
Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...
- 6,853
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PDF of the summation of L lognormal RVs
Given the following summation
$$\gamma = \sum_{l=1}^{L} y_{l},$$
where the PDF of $Y$ follows the lognormal distribution and is given by
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{...
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91
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Asymptotic moment of a multivariate normal distribution
Let the pdf of a multivariate normal distribution be
\begin{equation}
p_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2).
\end{equation}...
- 550
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86
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If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$?
Let $E$ be a separable $\mathbb R$-Banach space and $\lambda_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda_i(\{0\})=0$ and $$\int_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda_1-...
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315
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When is every Levy martingale of a process a continuous martingale?
Let $X_t$ be a real valued stochastic process, and $\mathcal H_t$ the the natural filtration of $X_t$.
Under what conditions on $X$ does the following statement hold?
For every $\mathcal H_\infty$-...
- 6,223
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479
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Probabilistic interpretation of derivative of a Dirac delta function
Consider $g : \mathbb{R}^d \mapsto \mathbb{R}$ defines some surface $\Sigma$ in $\mathbb{R}^d$. Then I can define a random variable $X_1$ with support only on $\Sigma$ by using a pdf of the form $$p_1(...
- 333
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301
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A monotonicity formula for the stochastic integral with respect to Brownian motion
Suppose $f, g: \mathbb R \to \mathbb R$ are continuous, non negative functions with $f \leq g$.
Fix some $T > 0$, and denote by $X^f$ the stochastic integral $\int_{[0, T]} f(s) \ dW_s$, where $W_s$...
- 6,223
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2
answers
616
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Optional stopping theorem: different versions
$(X_k)_{k \in \mathbb{N}}$ is a sub-martingale, $R_1,R_2$ two stopping times such that $R_1 \leq R_2$.
The optional stopping theorem which I know is the following (from probability theory, ...
- 249
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86
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Is integration against an indicator Wasserstein-Continuous
Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...
- 5,405
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66
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Convergence of an orthormal expansion of the density
Suppose that $X_1,..,X_n$ are i.i.d real random variables with density $f \in L_2(\mathbb R)$, and that $g_i$ are function forming an orthonormal basis of $L_2(\mathbb R)$, i.e :
$$f(x) = \sum\limits_{...
- 686
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184
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Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013)
On page 2 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the ...
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268
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Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?
From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem:
Tightness: Let $\Pi$ be a ...
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2
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874
views
Bounds for the sum of dependent gaussian random variables
Let $X_1,...,X_n$ be $n$ gaussian random variables $N(0,1)$ not necessarily independent or jointly correlated, $S=\sum_{i=1}^n w_i X_i$ be the weighted sum of these gaussian variables (because $(X_i)_{...
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Independence under regular conditional probability
Let $\{X_j\}_{j=1}^{n}$ be a independent identically distributed random variables taking values in $\mathbb{R}^d$. We write $\mu$ for the distributions. We assume moreover that $\mu$ is absolutely ...
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Distribution of line segment intersections in random pointsets
let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le ...
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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) ...
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Prove the the interval of selected elements in a list is exactly 4 [closed]
Assuming we have a ordered list of 115 elements, if we select 60 elements from the list, prove that the interval for at least 2 selected elements are exactly 4 elements apart.
Example:
...
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Weak convergence to a "multi-Bernoulli" distribution
Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming ...
- 449
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1
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201
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What is the most likely sequence? [closed]
I have a jar containing n numbered marbles, where 1...x marbles are red and marbles x+1...n ...
- 117
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1
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150
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Expectation of random matrix
Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that
\begin{align}
E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\...
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Number of duplicate pairs in multiple samplings
My universe has M different items. I run m=10 independent samplings over M. In each sampling, n elements are picked without replacement (n<<M). What is the expected number of pair duplicates we ...
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217
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Independent sampling of dependent random variables
Let $x_1, \ldots, x_n$ be possibly dependent random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. ...
- 153
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428
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First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
- 1,467
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2
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174
views
Asymptotic properties of ANOVA when the number of groups goes to infinity
Suppose
$$X_{ij} = \mu_j + \varepsilon_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N_j$$
ANOVA can allow us to test whether $\mu_1 = \cdots = \mu_J$.
In traditional ANOVA, however, the number ...
- 331
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2
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246
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Finding the expectation of $a \mathcal{Q} \left( \sqrt{b } \gamma \right) $, where $\gamma$ is a Gamma r.v
I'm trying to analytically find the following expectation
$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right],$$
where $a$ and $b$ are constant values, $\mathcal{Q}$ is the ...
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1
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57
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Expected size of binomial coefficient with Poisson arrivals?
I have a Poisson process where new elements arrive to a set with Poisson intensity $\lambda$. Initially, there are $N_0$ elements in the set. The probability that there are $N_0 + M$ elements in the ...
- 371
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78
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Estimate an expression about probability about Bernoulli random variables
Given $v_{ij} \in \{0,1\}$, $i \in \{1,2\}$, $j \in \{1,2,\ldots,n\}$. Let $X_1, X_2, \ldots, X_n$ be random variables, $P[X_i=1]=P[X_i=0]=1/2$, $i \in \{1,\ldots, n\}$. By checking many examples, I ...
- 6,201
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215
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Conditional distribution for the random walk on $\mathbb{Z}$
Define a random walk on the integers $\mathbb{Z}$ with step distribution $F$ and initial state is zero which is a sequence $S_n$ of random variables and its increments are iid random variables $X_i$ ...
- 288
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123
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Is there a proper name for those 'shifted moments'?
Suppose that we have a random variable $X$, $i \in \mathbb N$, and a scalar $t$. Is there a proper name for these integrals, that I for the moment call 'shifted moments' ?
$$I_{i}^{t} = \mathbb{E}\...
- 686
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1
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684
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The properties of total variation metric
The questions was asked by me on Math StackExchange, but no answer appears, so I ask for help again.
Let $(X, d)$ be a complete (Hausdorff, separable, local compact and other nice properties you want)...
- 1,799
0
votes
1
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694
views
expectation of the trace of the square root of wishart matrix
Let $X(N,N)$ be Wishart matrix with rank(X)=K in order to estimate the expectation of the trace of the square root of X i.e $X^{1/2}$ I want to know if is possible to use the unordered Wishart ...
- 377
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2k
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Martingale convergence theorem in Polya's urn
I want to get checked if my attempt is okay.
First off, let me shortly describe what Polya's urn is:
A certain urn initially contains a red and a blue ball. We now repeatedly do the following : we ...
0
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1
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66
views
Minimum mean over all random variables subject to logarithm constraint
Does the following problem have a solution?
$$
\min_X \mathbb{E} X
\quad\text{subject to}\quad
\mathbb{E} \log X = C.
$$
Here, the minimization is with respect to all integrable random variables $X$ ...
- 3
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1
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1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
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1
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378
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Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
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208
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Local behavior of the Vandermonde convolution
An interesting combinatorial identity is the Vandermonde convolution identity:
$$ \sum_k {n\choose k}{m\choose s-k} = {n+m \choose s},$$
which can be proved by considering the coefficients in $(x+1)^{...
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