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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44 views

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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20 views

minimization of a function with random variables

Let $x_1$ and $x_2$ two random variables and $f_{(x_1,x_2)}$ a positive function that depends on ${(x_1,x_2)}$. I want to minimize $f_{(x_1,x_2)}$. My question is which of these two approches is ...
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1answer
30 views

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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1answer
52 views

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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14 views

Solving an implicit equation with random parameters

I came across the following implicit equation for a real quantity $X$: $$G(X) = X - 1 - \dfrac{a}{n}\sum_{i=1}^n \dfrac{1}{X - b_i} = 0$$ where $a$ is a constant and the $b_i$ ($i = 1,\dots, n$) are ...
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16 views

Integral analysis of random variable

Need help in solving the $Pr(\frac{xy+ayz}{xz+ayz}<b)$ where x,y,z are exponential random variables with parameter $\alpha$, $\beta$, $\delta$ and a and b are constants. How do I solve this ...
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1answer
56 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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1answer
42 views

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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47 views

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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12 views

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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38 views

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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15 views

Total variation distance doenst increase when applying function

I am looking for a proof about the total variation distance that if for two independant variables $U,V$ holds $|P(U)-P(V)|_{TV} \leq \epsilon$, then also for every function $g: |P(g(U))-P(g(V))|_{TV} \...
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18 views

Does the central limit theorem imply that the sum of many independent random fields is a Gaussian random field?

The central limit theorem implies that the sum of many independent, random variables with finite variances approximates a Gaussian. Does it also imply that the sum of many independent, random fields ...
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18 views

Modulus of continuity for Random Functions

Let $G$ me a metric space and $T$ be an index set. Let $s_t(x): T \times G \rightarrow \mathbb{R}, t \in T \text{ and } x \in G$ be a spatio-temporal, determinstic field continuous in $G$ for all $t$. ...
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26 views

Rotationally invariant random matrices

Consider a $2n-$dimensional random matrix $M$ of the form, $M = \begin{bmatrix} aa^T & ab^T \\ ba^T & bb^T \end{bmatrix}$ where $a$ and $b$ are $n$ dimensional random vectors. Are there any ...
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16 views

Uniform distribution with sampled function

A processes $P$ is started at $t=0$. We denote their running time, the difference between the instant they stop and $t$, as $r$. We denote the maximum running time of the process as $T$. The process ...
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2answers
49 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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2answers
43 views

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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9 views

Question on random sampling of e periodic signal

Suppose we have a function $$ D(t) = \text{const.} + \tilde{D}(t) $$ with $$ \tilde{D}(t) = \sum_{k=1}^K a_k \, \sin(\omega_k t + \phi_k) $$ with $a_k, \omega_k, \phi_k \in \mathbb{R}$ . Let $$ \...
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10 views

How are deterministic and random dimensions of a random function (a r.v. function of another r.v.) reconciled w.r.t. the function's output value?

I am struggling to reconcile the existence of the deterministic and stochastic dimensions of Y defined at the same time. Specifically, if X, Y are r.v.'s and Y is a function of X s.t. Y = g(X), where: ...
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1answer
62 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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1answer
47 views

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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17 views

PDF of function of a random vector

I have a random vector $\vec x \in R^{n}$ and its pdf $f_\vec x(\vec x)$ and another vector $\vec y \in R^{m}$ with pdf $f_\vec y(\vec y)$ where $n > m$. A non linear function $g(): R^{m} \...
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2answers
46 views

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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12 views

For $z \sim N(0,(1/m)I_m)$ and fixed $a \in R^m$, compute $\sup_{x \in R^m,\;\|x-a\| \le r}\mathbb P(f_z(a)f_z(x)\le 0) $, where $f_z(x):=z^Tx$

Let $m$ be a large positive integer and let $z = (z_1,\ldots,z_m)$ be a random vector in $\mathbb R^m$ with iid coordinates from $N(0,1/m)$. Consider the random function $f_z:\mathbb R^m \to \mathbb R$...
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17 views

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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1answer
51 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 ...
2
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1answer
41 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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16 views

asymptotic normality of iterative lipschitz mapping: $X_{n}=f(X_{n-1})+w_n $

I am interested in the following problem: Consider the sequence of r.v. ${(X_n)}_{n=1}^{\infty}$ defined recursively as $$X_{n}=f(X_{n-1})+w_n $$ where $f$ is some fixed and known nonlinear lipschitz ...
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4answers
84 views

Expectation of $Y=X^{6}$ : $X \sim \mathcal{N}(0,1)$?

Using LOTUS: $$\text{E}(Y) = \int_{-\infty}^{\infty}dx \Big( x^{6} \cdot \frac{1}{\sqrt{2 \pi}} \text{exp}(-\frac{1}{2}x^{2}) \Big)$$ I have two functions multiplied inside the integral, so use ...
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1answer
89 views

Probability distribution for a function of random variables

I'm very new in the Statistic Math field, so this question maybe be a bit trivial for you guys. Anyway, I'd appreciate any guidance in this matter. I was thinking about whether is possible to find the ...
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0answers
18 views

Probability of similarity for two random time series

I was looking for the method to calculate theoretical similarity of two random noises $x$ and $y$ with mean $0$ and standard deviation $1$, I've got also $12500$ samples of signal, sampled $2.5$ Ga/s. ...
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0answers
14 views

What will be the sum of two Complex gaussian mixture $x$ and $y$?

I have two distribution known as Gaussian mixture with 2 and 3 complex Gaussian component respectively such as, $x = \sum\limits_{i = 1}^2 {{\varepsilon _{x,i}}CN\left( {{\mu _{x,i}},\sigma _{x,i}^2} \...
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1answer
36 views

Convergence in law for a family of random generalized functions

Let $\Phi_a$ be a family of random generalized functions all belonging to the same Sobolev space with negative exponent, say $H^{-k}$ for $k>0$. Is it possible to speak about convergence in law in $...
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0answers
20 views

Literature request — uniqueness and existence of a specific type of ODE

I am looking for a proof of the existence and uniquenes of ODE's of the type: \begin{equation} \dot{f}(t,x,y) = F(h(t,x), f(t,x,y)), \end{equation} where $f : T \times X ...
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0answers
18 views

Is there a simple reduction from permutations of {1, … 2M} to {1, … M}?

Suppose I have a random permutation uniformly chosen from the set of all permutations of $M N$ elements; often in the contexts that I am interested in, this is going to be $2^{m+n}$ elements. And let ...
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0answers
7 views

Continuous variable function that evaluates to random curves

There exists a function of a continuous parameter function that sometimes evaluates to one curve and other to a different one? For instance, is there a function $f(x)$ that say is 0 for $x < 0$ and ...
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0answers
8 views

Bound the error for approximating $g(Z) \approx g(h)$, where $Z \sim N(h, \sigma I)$ and $g$ is high-dimensional

I have multivariate Gaussian RVs that have the following properties: $Z \sim N(h, \sigma_Z I)$, where $h$ is a deterministic vector, $I$ is the identity matrix. $X = g(Z) + N_x$, where $g$ is a ...
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1answer
23 views

Relationship (sum/difference) between elements having normal distribution

I have a set of elements $\{x_1, x_2,...,x_n\}$ having normal distribution. Now, I want to choose $K$ random pairs $(x_i, x_j)$ and compute their difference $\Delta x = |x_i-x_j|$. How can I compute ...
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2answers
46 views

Expected value of a transformation

If $X$ is a continuous random variable with $EX = \mu < \infty$ and $Y = \exp(a|X|)$ for some $a > 0$ Is $EY < \infty$. How might one go about confirming this? Is knowing the distribution ...
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2answers
66 views

Can a function that selects between two random variables increase the variance more than twofold?

Let $X_1,X_2$ be two real-valued zero-mean random variables, and assume w.l.o.g. that $\text{Var}[X_1]\ge\text{Var}[X_2]$. Let $f:\mathbb R^2\to\{1,2\}$ be a ``selection'' function, and define $Y=X_{...
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0answers
98 views

Deconvolution of a mean-preserving spread

Context I have been working on proving the existence of a mathematical object. After trying several things, I think that if I can show the following, an important step towards proving existence will ...
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1answer
180 views

Convergence in Lp implies almost sure convergence of a subsequence

Assume that $f_n$ is a convergent sequence in $L^2(\Omega, L^2(0, 1))$ with limit $f$. Is it true that there exists a subsequence $f_{n_k}$ converging almost surely, i.e. such that $$ f_{n_k}(\omega) \...
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0answers
27 views

Determine pdf of a random process given a fixed parameter

Given a random process $X(t) = Y \cos(\omega t)$, where $Y$ is a uniform RV on $[0,1]$, $t \geq 0$ and $\omega$ is a constant. Determine the pdf at $t = \frac{\pi}{2}$. For any other given $t$ I ...
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3answers
470 views

Algebra to pick random element from a set

Lets define set $G$: $$G = \{ 1,2, \dots,n \space | \space n \in \mathbb{N} \} \text{ and }\mathbb{N} \rightarrow \mathbb{R}_+$$ What is the algebraic notation to build set $Y$ by picking randomly 20 ...
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2answers
72 views

probability function of a function of random variables

For any random variable that is a function of other random variables, e.g. $Z = g(X_1,X_2,X_3) = 5X_1 + (X_2X_3)^2$, is there a general formula/method to find $f_Z(z)$ given one knows $X_1,X_2,X_3$ ...
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0answers
37 views

Central Limit Theorem for m-dependent stationary random variables in $L^2[0,1]$

I am currently writing my master thesis and have a problem with Lemma 2.5 in http://www.math.utah.edu/~rice/berkeshorvathrice2012.pdf . I have to show that $$\frac{1}{\sqrt{N}}\sum_{i=1}^{\lfloor Nx \...
2
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1answer
49 views

Multiplication of random matrices with independent entries

Let A and B be two random matrices with zero-mean i.i.d. entries. Then, are the entries of C = A*B independent? From intuition, each entry of C is the dot product of two different independent random ...
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1answer
218 views

How to find the probability mass function and mean of a function Y of binomial random variable X.

So, suppose I have a binomial random variable $X$ with parameters $n=6$ and $p=(1/5)$. Now, if I have a function of $X$ like $Y=(X-2)^2$, how do I go about finding it's properties - and in particular, ...
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0answers
50 views

Why is Knuth's definition of random sequence (R4) too weak?

Knuth's $R4$ definition of a random sequence is called "too weak" and he presents an example of why --- second paragraph after the definition on the previous link. Definition $R4$. A $[0..1)$ ...

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