Questions tagged [random-functions]

This tag is for questions relating to the functions of random variables which is a function from $Ω$ into a suitable space of functions (where $Ω$ is the sample space of a probability space that has been specified). Technically, there is also a measurability condition on this function.

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Characteristic function of the mean of a Dirichlet process

In a 1984 paper discussing the characteristic function of the mean of a random distribution driven by a Dirichlet process ${\sf DP}(M,G_0)$ (Ferguson, 1973), $M>0$, Hajime Yamato sets a constraint ...
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1 answer
42 views

Events of correlated jointly Gaussian distributions

Suppose you have $X,Y\sim\mathcal{N}(0,1)$ jointly Gaussian distributed with correlation coefficient $\rho$. I am looking for a convenient formula of the following expression: $$\mathbb{P}(\{|X|>1\}...
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Does $P(X+Y=k | Y = k) = P(X = 0)$ hold true?

Given two independent random variables $X$ and $Y$. Does the statement $P( X + Y = k | Y = k) = P(X = 0)$ hold true always? My logic is $P( X + Y = k | Y = k) \\= \frac{P(X + Y = k \cap Y = k)}{P(Y=...
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Is indistinguishable probabilities still indistinguishable even if randomness is allowed?

We say that a function $f:\mathbb N\to\mathbb R$ is negligible if for all positive integer $k$, there is $N\in\mathbb N$ such that for all $n\geq N$, \begin{equation} |f(n)| < \frac{1}{n^k}\...
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Size of preimage in a pseudo random function

I am interested in statistics of pseudo random functions. In particular, in the following. Given: a pseudo random function $f : S \to S$ (with $s = |S|$), and a set $D \subseteq S$ (with $n = |D|$) ...
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pdes describing posterior mean and covariance functions of a gaussian process

In the paper titled "Stochastic processes in several dimensions", Whittle showed that matern gaussian processes satisfy a certain spde. Does this then imply that the posterior mean and ...
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1 answer
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Picking a special function from the set of random functions

Consider a fixed integer $q$. Consider the set of all functions from $\{0, 1\}^{n+1}$ to $\{0, 1\}^{m}$. Let us pick one function from this set uniformly at random. Now, let's say we want functions $f$...
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Is a function on a product space measurable if it is equal almost surely to a product measurable function?

My question is the following: Consider Borel spaces $(S, \mathcal{S})$ and $(T, \mathcal{T})$, where $\mathcal{S}$ and $\mathcal{T}$ are Borel $\sigma-$algebra of $S$ and $T$ respectively. Consider a ...
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1 answer
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Having problems in understanding this multivariable problem example

So the problem says: Inside interval [0,1] dot (a) is fixated. Random variable X is uniformly distributed on interval [0,1]. What is the covariance moment between X and variable Y = |x-a| : distance ...
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Posterior distribution of parameters for a dynamical system given data

I am stuck trying to derive an expression in the context of Bayesian inference for a model inverse problem. Specifically, I am considering a dynamical system of the form $X(t)=\Phi(\theta,t)$ and ...
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Random process Mean function and Correlation function

Suppose $Z(t)=Σ_{k=1}^{n}Xe^{j(𝜔_0t+𝚽_k)}$, $t∈R$ where $𝜔_0$ is a constant, $n$ is a fixed positive integer, $X_1,...,X_n, 𝚽_1,...,𝚽_n$ are mutually independent random variables, and $EX_k=0, ...
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1 answer
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If $X_t = Y_t$ in distribution, for any $t \in T$ (compact), is it true that $\mathbb E \sup_{t \in T} X_t = \mathbb E\sup_{t \in T} Y_t$?

Let $T$ be a compact topological space and for any $t \in T$, let $X_t$ and $Y_t$ be random variables which have the same distribution. We may assume that $X_t$ and $Y_t$ depend on $t$ in continuous / ...
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Autocorrelation function of a random process calculation

A random process X(t) is expressed as: X(t) = 5 cos(500πt + φ) + W(t), where the phase (φ) is uniformly distributed over the interval [−π,π]. In addition, the process Gaussian random W(t) is white ...
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1 answer
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Probability density function of absolute sum of normal and uniform random variables

Description of context Given are independent random variables $n,u$ that are normally and uniformly distributed, $$n\sim\mathcal{N}_{\mu,\sigma}=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\left(-\frac{1}{2}\...
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1 answer
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Uniformly simulating random functions with derivative bounded by fixed constant

I want to be able to uniformly draw (finite approximations of) functions $f: [0,1] \to \mathbb{R}$ such that $f(0)=a$, $f(1)=b$, and $|f'(x)|<s$ (for a fixed $s$). I want to do this so I can draw ...
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If Z=X+Y for random variables X and Y, can I transform X and Y once I derive the distribution of Z?

I am trying to derive the distribution of $Z = X + Y$, where $X$ and $Y$ are normally distributed and but not necessarily independent random variables. To make the math easier, I started by ...
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Under what conditions will the variance of max(X,Y) be greater than max(X-Y,0), if X and Y are random variables?

This is a bit of an open-ended question that's been bugging me for a while, and any help or insight would be appreciated. My apologies in advance if I make any math sins, please correct me if so. ...
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1 vote
1 answer
32 views

Measurability of (random) set valued functions

Consider the following problem. Given a set $A\in \mathcal{B}(\mathbb{R})$, we have the associated indicator function $1_A(x) \in L^\infty(\mathbb{R})$, is this mapping, $A\mapsto 1_A$ in some sense ...
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Is it possible for me to derive a function that uses a random integer to derive a pseudorandom number from an indexed list of sequential numbers?

Assuming I have one random integer, let's say 123456, and an indexed list of numbers [1, 2, 3 ... 100] Is it possible for me to derive a function that uses the random integer to derive a pseudorandom ...
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How can we demonstrate that the local variance tends to increase with local mean for a random field obeying lognormal distribution?

In the classic geostatistics book "Goovaerts,P., 1997. Geostatistics for natural resources evaluation. Oxford University Press", it said that "For positively skewed distributions, the ...
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Whether the autocorrelation structure of random field $Z(u) + Y(u)$ is equal to the autocorrelation structure of $Z(u)$ plus that of $Y(u)$?

I want to simulate a Gaussian random field (RF) with correlation structure (represented by the geostatistic tool 'semivariogram' $\gamma (h) \: +\: pure \: nugget \: effect$). I want to know whether ...
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2D space and 1D time evolution of a random field

I want to develop a 2D random field and its change with time with constant velocity. My process: Define a 2D grid (not the fields yet) $[x, y]$ with $n \times n$ points Define 1D time axis $[t]$ with ...
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Maxima/minima of a random curve

Let $\left\{ {{\xi _k}} \right\}$ be independent random variables with some known distribution function ${F_\xi }$ and $f(x)$ be a "good" function that is bounded and decrease to zero with ...
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1 answer
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What is the expected fraction the domain of a random function that maps the function to a certain interval?

Consider a random function $\phi(\underline{x})$, where $\underline{x}$ is a spatial position. For each independent realization $\phi^{(k)}(\underline{x})$ of the function there is a certain fraction $...
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Where can i find collection of Problems on Transformations of random variables

I am looking for a book or a website or any other source where i could practice questions on finding PDF of $Y=g(X)$ where the PDF of the random variable $X$ is given. Any inputs for this ?
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Non-iterative deterministic function to map an input to a random output in range without repeating

Apologies if this has been answered before or is impossible, but: Is there a state-independent, non-iterative function that, given an (integer) input (n) and (integer) minimum (min) and maximum (max) ...
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What is the Borel sigma-field of positive functions?

I was reading the paper "The Logistic Normal Distribution for Bayesian, Nonparametric, Predictive Densities" published in 1988 in Journal of the American Statistical Association by Peter J. ...
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How can I obtain the CDF of a function of two variables

Consider two nonnegative random variables, $X$ and $Y$. Assume that their PDF, i.e., $f_X(x)$ and $f_Y(y)$ are given. Assume that an arbitrary function $g:X\times Y\to\mathbb{R_{\ge0}}$ is given. ...
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Conditions for the convergence of the argmin of two random functions over a random set

Suppose I have a sequence of continuous functions $f_n, g_n \colon S \to \mathbb{R}$, where $S$ is a compact subset of $\mathbb{R}^n$, such that \begin{equation*} \sup_{x \in S} | f_n(x) - g_n(x) | \...
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Algorithm: Randomize Brightness and Contrast Values, With Constraints

I want to do a random brightness and contrast adjustment to an image, such that... ...
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Random spreading control by math functions

I want to take control of random spreading by math functions. Here is my try with formula: $|R^{|P|}| \cdot \text{Falloff}\cdot \operatorname{sgn}(R) + \text{Offset}$ $R$ is random number between $-1....
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6 votes
2 answers
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Linear-time sampling of stochastic processes?

Are there any stochastic processes $(X_t)_{t \in \mathbb{R}^d}$ such that almost surely paths are continuous but nowhere differentiable and sampling of $n$ points $X_{t_n}$ on a path can be done in $...
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1 vote
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How do I prove a sequence is disjunctive?

I wrote a random number generator with an unbounded state size. I don't know where to begin proving it to be (or proving it isn't) disjunctive. What would be a property of a disjunctive sequence, ...
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1 answer
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limit of sequence with $o_{p}()$

I am confused with sequence where $o_{p}$ is involved. Assume we have the following sequence of random variables, defined on the same probability space: $$ x_{n} = \frac{M_{n}+a}{M_{n}(M_{n}+b)}, $$ ...
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Convergence for a Random Normal Process defined by Recursion

A question from my Random Processes exams: Let $ W_0,W_1,W_2,...$ be a sequence of independent Gaussian random variables with Mean 0 and Variance $ \sigma ^ 2 > 0 $. Define the sequence $ (X_n : n \...
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4 votes
1 answer
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The weak limiting point of a stationary random field is stationary.

Let $(\Omega,\mathcal{A}, \mu, (\tau_x)_{x \in \mathbb{R}^3}$) be a probability space endowed with an ergodic dynamic system $(\tau_x)_{x \in \mathbb{R}^3}$. We say that a function $\phi :\Omega \...
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1 answer
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Expected value and variance for a homogeneous polynomial of random variables

I want to obtain equations for expected value and variance for a homogeneous polynomial of arbitrary order $n$. For a quadratic form, which is a special case of a homogeneous polynomial, there are ...
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1 answer
197 views

Probability distribution of a function of two random variables

I'm trying to find the probability distribution of a function $q(x, y)=x^2+y^2$, where $x$ and $y$ are normally distributed: $x$~$N[0,\sqrt{V_x}]$ and $y$~$N[0,\sqrt{V_y}]$. I began by finding the ...
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Highly Random Function

Call a function $f\colon\mathbb{R}\to\mathbb{R}$ highly random if: Say $T$ is a Turing machine which attempts to compute values of $f$. Given enough values to compute, the cumulative error of $T$'s ...
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optima of a random function / stochastic process

Consider the probability space $(\Omega, \mathcal{F},\mathbb{P})$, and let $X:\Omega \mapsto \mathbb{R}^T$, where $T$ is an index set, be a random function. What is the canonical definition of the ...
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Is it possible to manipulate a random stream?

I got two random streams of binary data $A$ and $B$. During each step, I can choose to pull a byte from stream $A$ or stream $B$. My goal is to reduce the chances of pulling a pattern* $P$. Is there a ...
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How to compute E[$g(X)$] in terms of $X$'s CCDF (complementary cumulative distribution function)?

If we know the PDF $f$ of a random variable $X$ then we can compute an expression like $\mathrm E[g(X)]$ as $$\mathrm E[g(X)] = \int_{\mathrm{Im}(X)} g(x) f(x) \mathrm dx \, .$$ Let $F$ be the CDF of ...
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1 vote
2 answers
208 views

Probability of generating a sequence of numbers between 1 and n

Let's say we want to generate a random number between 1 and n, n-times,then the probability that every integer between 1 and n appears once in the generated sequence of random numbers is $\frac{n!}{n^...
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2 answers
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Joint density of functions of two random variables

If $X$ and $Y$ are iid with $U(0,2)$, $Z=2X+Y$ and $V=e^X$ What is the joint density of $(Z,V)$? $\begin{align} V=e^X &\Rightarrow X=lnV\\ Z=2X+Y &\Rightarrow Y=Z-2lnV \end{align}$ $ J= \begin{...
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181 views

Probability of $k$ fixed points for a random function from and to $\{1,..,n\}$

I would like to derive the probability mass distribution $p_k$ of the number $k$ of fixed points of a random function from $A:=\{1,..,n\}$ to the same set. I proceed computing the number of ...
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Reversing an LCG

I'm having a hard time finding an answer to this. I've found several places that discuss it, but they do a very poor job of helping me (specifically me, maybe I'm dumb) understand what they're doing. ...
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2 answers
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Probability that quadratic polynomial in independent gaussian variables is negative

Let $a,b \in \mathbb R^m$ be fixed (deterministic vectors), and let $w=(w_1,\ldots,w_m)$ be a random vector in $\mathbb R^m$ with iid coordinates from $N(0,1)$. Define the random variable $$ h(w) := w^...
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1 vote
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Use Kac-Rice to find expected number of zero-crossings of the process $X_t := \alpha |Z_1| - t\|Z\|^2/m$, $Z \sim N(0, I_m)$

I'm trying (hard) to understand the Kac-Rice theory for computing the expected number of zero-crossings of a random process. To this end, I've identified a simple problem which I know how to solve ...
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  • 8,289
2 votes
1 answer
140 views

Relationship between autocorrelation function and wavelet coefficient

the autocorrelation function can be represented using the spectral density in Fourier space. Is there a similar relationship between the autocorrelation function and the coefficient in the wavelet ...
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2 votes
1 answer
70 views

An Application Kolmogorov's Three Series Theorem

I want to prove the following question, which is found in this practice exam: My attempt so far is as follows - I just can't show that the $\sum E(Y_i)$ converges.
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