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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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Distribution of difference of two random variables

The problem is following: Suppose we have a line segment with a length $L$ and we randomly choose (with uniform distribution) $n$ points on this segment, so we divide the main line segment into $n+1$ ...
XaveryXavier's user avatar
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18 views

Function of exponential random variables

I have $20$ exponential random variables with mean $\alpha$ representing delays $D_n: n \in\{1,\ldots, 20\}$. I have 20 random variables denoting power $P_n^{'}: n \in\{1,\ldots, 20\}$. which depends ...
wanderer's user avatar
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1 answer
113 views

What is the limit probability an element of $x \in S$ belongs to $f^n(S)$, for $n \to \infty$?

Let $S$ be a finite set of $|S|=n$ elements and $F$ be the set of all functions $f:S\rightarrow S$. It's easy to demonstrate that the integer sequence $\{c_i\} = |{\rm Im}(f^i)|$: is non increasing; ...
Yuri S VB's user avatar
2 votes
1 answer
57 views

Randomly Generating Real-Rooted Polynomial Equations

I need a simple function to generate real-rooted polynomial functions to demo my Desmos Aberth-Ehrlich rootfinding implementation. My current function is as follows: Let $n \in \mathbb{Z}^+$ be the ...
James Baw's user avatar
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15 views

Random functions and neuronal network

Let f be a continuous function on $\mathbb{R}$ and B_t a brownian motion. Is there any density result of the neuronal network class for function of the form $t \mapsto f(B_t)$ ?
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1 vote
2 answers
70 views

Birthday problem: how to show the scaling with $1/N^2$?

Suppose there is a sequence of$N$ numbers $x_1, x_2, x_3, ... x_N$. There are then gaps $|x_i - x_j|$, and the minimum gap: $\delta (N) = \text{min}_{i \ne j \le N} \{ | x_i -x_j | \}$. Let the mean ...
Nigel1's user avatar
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Keener Lemma 9.1 proof

I'm reading the book Theoretical Statistics by Keener, and I couldn't figure out one of the claims in the proof for Lemma 9.1. Lemma 9.1 states: let $W$ be a random function in $C(K)$ where $K \subset ...
statstats's user avatar
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56 views

How to prove that a set of mathematical operations is actually a pseudo-random permutation?

The area of PRNG is mostly an experimental area. I have created a simple PRNG (called komirand) which passes statistical tests for randomness in ...
aleksv's user avatar
  • 301
1 vote
1 answer
103 views

How does an integral change the distribution of a random variable?

Suppose I have a random variable $x$, and I want to perform the integral of a function of $x$ such that: $$y=f(x)=\int_{c_l}^{c_u} f(x,c) dc$$ where $f(x,c)$ is a nonlinear function of $x$ and $c$. ...
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Does distribution of input with a given entropy matters to the entropy at the output of a random function?

For random functions with $k$ input and output values, we can define the expected Shannon entropy of the output when the input has a uniform distribution (with the expectancy over all random functions)...
fgrieu's user avatar
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Convergence of expectations of bivariate functions

Suppose I have a continuous bivariate function $g:\mathbb{R}^2\rightarrow [0,\infty)$ that is increasing in both arguments and uniformly bounded by some constant. I also have three sequences of random ...
ORgeek 's user avatar
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Asymptotic integral over rapidly randomly oscillating function

As a part of my bachelor thesis on cosmic structure formation I have been dealing with a sum of many randomly distributed phase factors, so in principle with a pearson random walk. If there are $N$ ...
Ricardo Ochel's user avatar
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28 views

central limit theorem for random field

Let $x\in(1,\infty)$ and let $Z(x)=\sum_{i\ge1}Z_i(x)$ be a sum of independent random variables $Z_1(x),Z_2(x),\dotsb$ such that each random variable is bounded as $Z_i(x)\in[-1/i,1/i]$. Moreover, it ...
amanwithnoname's user avatar
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Expected Value of Rewards from Coin Flipping

I am struggling with coming up with an approach for the following question (pulled from Assignment 4 from MIT's 6.041 Probability course, which I'm trying to self-study as I realized my probability ...
Anish Ganti's user avatar
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2 answers
79 views

Expected value of normalized random variable [closed]

I have $N$ random variables $X_1, \dots, X_N$ that are independent and identically distributed. Define the quantity $y_i = \mathbb{E}\left[ \frac{X_i}{\sum_{j=1}^N \alpha_j X_j}\right]$, where $\...
Chao's user avatar
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Difference between the random variable $f(x)\ge a$ for any $x$, and $\sup f(x)\ge a$

If our goal is to derive bound on $P(\sup_{x\in T} f(x)\geq a)$, where $T$ is a uncountably infinite subset $[0,1]^n$ of $\mathbb{R}^n$, and $f(x):=\sum_{i=1}^n 1+w_i$ ($w_i$ is gaussian scalar), then ...
happyle's user avatar
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Autocorrelation and Power Spectral Density for Wide-Sense Cyclostationary Processes

I am reading in a book called "Understanding Jitter and Phase Noise" and came across the following equation and need a little help to understand his justification for a certain step in the ...
Mohamed Osama's user avatar
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77 views

Using aperiodic monotiling for repeatable pseudorandom number generation

As a caveat, i am not a mathematician but rather a programmer with an amateur interest in patterns, fractals, sequences, data science. That said, i have been following recent developments in aperiodic ...
simonalexander2005's user avatar
1 vote
1 answer
107 views

Relation Between Subgradients of a Random Function and Its Expectation

Suppose $\mathcal{X}\subset\mathbb{R}^n$ is a convex set. Let $f:\mathcal{X}\times\mathbb{R}^m\to\mathbb{R}$ be a function such that for every $y\in\mathbb{R}^m$, the function $f(\cdot,y)$ is convex, ...
Kittayo's user avatar
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1 answer
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Is there any literature on union of random mappings?

I have studied the landmark papers of Rubin et al, Harris, Flajolet on Random mapping statistics. I have also read some follow up papers. They provided analysis of the structure of random maps. My ...
Bishwajit Chakraborty's user avatar
2 votes
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43 views

The distribution of the composition of two random functions

Let $A$ and $B$ be two random functions of binary variables (e.g., two probabilistic algorithms). Then, on input $x \in \{0,1\}^*$, $A(x)$ outputs $y \in \{0,1\}^{f_A(|x|)}$ with probability $p_A(A(x) ...
trillianhaze's user avatar
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1 answer
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Difference between $P(f(x*,w)>0)→1,P(f(x,w)>0)→1$ and $P(min(f(x*,w),f(x,w))>0)→1$ when dimension grows

Let $f(x_1,\cdots,x_n,w)$ be a function from $R^{n+1}\rightarrow R$, where $x_1,\cdots,x_n$ are deterministic variables, and $w$ be random variable. As a simple example, $f$ can be $(x_1+\cdots+x_n)w$....
happyle's user avatar
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Problem with distribution of $Y=X_1 X_2$ via known $f_{X_1,X_2}$ plus extension to arbitrary monomial function $Y=g(X_1, ... X_n)$

I hope you're well. Question 1 The first problem I have is essentially that I want to find the product distribution (CDF) of two random variables ($Y=g(X_1,X_2)=X_1 X_2$), who's joint PDF distribution,...
fincleah's user avatar
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2 answers
107 views

Variance of min of r.v. and constant

I am not a student of statistics, but need to compute an expression for my work. This is what I have so far: I have a r.v. $D$ (pdf: $f$, support: $[0,\infty]$), and a positive constant $q$. I have a ...
Richa's user avatar
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153 views

Application of Karhunen–Loève theorem to Wiener process

The Karhunen-Loève Theorem is concerned about a continuous second-order process $X=\{X_t, t\in [a,b]\}$, defined on a probability space $(\Omega,\mathcal A, P)$. The Theorem allows us to apply the ...
Celine Harumi's user avatar
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PAC learning in the convex case with bounded expected risk

In the context of computational learning theory: Let $f(w,z): \mathcal{W} \rightarrow \mathbb{R}$ be convex in $w \in \mathcal{W}$ where $\mathcal{W}$ bounded by R. Over $\mathcal{W}$, $|f| \leq C$. ...
Yakov's user avatar
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1 answer
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Identifying the number sequence

Once I needed to calculate a certain quadratic form on random vectors: $$ I = \sum_{i,j=1}^N x_i x_j \;, $$ where $x_i = \pm 1$ are random variables, the probabilities of their values are $P[x_i=-1] =...
Evgeny P. Kurbatov's user avatar
1 vote
2 answers
146 views

Continuity of sup-norm for random functions

I have a problem when reading through Theoretical Statistics by Robert. W. Kenner, Theorem 9.1 in Chapter 9, pp. 152-153. It is about the continuity of random functions. By random functions, they are ...
ムータンーオ's user avatar
-1 votes
1 answer
67 views

Generate six random numbers that come $1, 2, 3, 4, 5, 6$ with the given ${\tt PMF}$ according to a sequence $.1, .1, .2, .3, .2, .1$ using non-uniform

Problem. Generate six random numbers that come $1, 2, 3, 4, 5, 6$ with the given $\texttt{PMF}$ according to a sequence $0.1, 0.1, 0.2, 0.3, 0.2, 0.1$ using non-uniform random number generator. For ...
user avatar
0 votes
0 answers
29 views

Random process and statistics

Im trying to solve this and im really having trouble with understanding how to do this lets say given two random variables x1,x2 that are independent and uniformly distributed between 0 and 1 we are ...
Avi Bents's user avatar
1 vote
0 answers
54 views

What are functions taken from Gaussian random fields (grf)?

I am trying to use deep learning in order to learn the integral operator. For that I have to take a random function, such that I can apply the anti-derivative and generate my training dataset. I am ...
Formal_that's user avatar
1 vote
0 answers
288 views

What does a random continuous function look like?

My question may be a little strange, but I'm wondering how random continuous functions work? First, how is it possible to define a random continuous function and how to investigate it. For example, ...
oclelot335's user avatar
1 vote
1 answer
243 views

Sum of dependent random variables and copulas

I have two dependent continuous random variables (RVs) $X$ and $Y$ and I'm interested in determining the CDF of the sum, i.e., $F_{X+Y}(t) = \mathbb{P}(X+Y \leq t)$. I know the marginal of $X$ and $Y$ ...
Jeremy's user avatar
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1 vote
1 answer
132 views

What is the sum $\sum_{m} e^{i (U_m k + \beta_m)} $ when $U$ and $\beta$ follow different distributions

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ $i = \sqrt{-1}$ Here, $U_m$ are samples drawn from a Gaussian random distribution. $$ U_m \sim \mathcal{N}(\mu, \sigma) $$ ...
CfourPiO's user avatar
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1 answer
127 views

Covariance, and the Taylor expansion for the expected value of a linear function of random variables

Suppose I have two correlated random variables $X$ and $Y$ and am interested in the quantity $$ \theta = U(X) - U(Y)\ , $$ where U is a smooth function. I am trying to determine the correct Taylor ...
Anthony's user avatar
  • 813
0 votes
0 answers
150 views

Reverse Engineering a function based on inputs and outputs

I have a function that takes a whole number and outputs four values. For now I am only worried about matching the first 2 outputs. What is the best way to do this? I would use regression but due to ...
Jay McArthur's user avatar
1 vote
1 answer
49 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\}...
courageousmartingale's user avatar
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0 answers
47 views

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=...
mortoman's user avatar
1 vote
0 answers
111 views

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 $n = |D|$) ...
doc's user avatar
  • 1,307
3 votes
1 answer
76 views

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 $f$...
RandomMatrices's user avatar
0 votes
1 answer
47 views

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 ...
curious's user avatar
  • 21
1 vote
0 answers
25 views

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 ...
Justin Feigelman's user avatar
1 vote
1 answer
61 views

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, ...
i_am_learning's user avatar
2 votes
1 answer
78 views

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 / ...
dohmatob's user avatar
  • 9,565
3 votes
1 answer
358 views

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}\...
granular_bastard's user avatar
2 votes
1 answer
21 views

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 ...
Nerif's user avatar
  • 168
1 vote
0 answers
88 views

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 ...
benb's user avatar
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0 votes
1 answer
503 views

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. ...
David Loungani's user avatar
1 vote
1 answer
99 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 ...
user2379888's user avatar
1 vote
0 answers
74 views

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 ...
Jay Welsh's user avatar
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