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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.

A function of an arbitrary argument $~t~$ (defined on the set $~ \mathcal{T}~$ of its values, and taking numerical values or, more generally, values in a vector space) whose values are defined in terms of a certain experiment and may vary with the outcome of this experiment according to a given probability distribution.

In probability theory, attention centres on numerical (that is, scalar) random functions $~X(t)~$; a random vector function $~X(t)~$ can be regarded as the aggregate of the scalar functions $~X_a(t)~$, where $~a~$ ranges over the finite or countable set $~A~$ of components of $~X~$, that is, as a numerical random function on the set $~T_1=T\times A~$ of pairs $~(t,a),~t\in ~T,~a\in~A~.$

For more details, find

$1.~$ "Theory of Random Functions" by A. Blanc-Lapierre & R. Fortet

$2.~$ "The theory of stochastic processes" by I. Gikhman

$3.~$ https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables

$4.~$ https://encyclopediaofmath.org/wiki/Random_function

245 questions
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'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
user3130920
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4 answers

Why is gradient noise better quality than value noise?

I have been reading about the mathematics behind Perlin noise, a gradient noise function often used in computer graphics, from Ken Perlin's presentation and Matt Zucker's FAQ. I understand that each grid point, $X$, has a pseudo-random gradient…
Joseph Mansfield
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13
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4 answers

Mathematical description of a random sample

Mathematical description of a random sample: which one is it and why? $X_1(\omega), X_2(\omega), ..., X_n(\omega)$, where $X_1, ..., X_n$ are different but i.i.d. random variables. $X(\omega_1), X(\omega_2), ..., X(\omega_n)$, where $X$ is a…
Leo
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11
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5 answers

Why do we need "perfectly" random numbers?

I periodically see articles about physicists or others coming up with a technique that generates a slightly more random number than was possible before, and how this is useful for encryption. But I've never see an explanation for why we need such…
Fred Hamilton
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3 answers

Projections of uniformly distributed $\mathbb{R}^3$ unit vector have uniform distribution

My question revolves around the following property: Let ${\bf u} \in \mathbb{R}^3$ be a random vector with uniform distribution on the three-dimensional unit sphere. Then the projection on any given unit vector $\bf v \in \mathbb{R}^3$ $$X = {\bf…
GDumphart
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8
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1 answer

Jensen's inequality for random functions in a Banach space

Suppose I have a Banach space of functions over $\mathbb{R}$ with norm $||\cdot||$. Suppose that $f$ is a random function that takes values in this space such that $||f||\leq 1$. Suppose that for all $x\in\mathbb{R}$ $E[f(x)]$ exists and is finite…
SecretlyAnEconomist
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What is the expectation of norm of $[X_1,\ldots, X_n]$ where $X_i$ are indpendent complex Gaussian random variables

Consider a random vector $X=[X_1, X_2, \ldots , X_n]$ where $X_i$ ($i \in 1, 2,\ldots, n$) are independent complex Gaussian random variables with zero mean and variance $\sigma_i^2$, i.e., $X_i \sim CN(0, \sigma_i^2)$. How can I find expectation…
Sabyasachi G
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7
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0 answers

Decrease of entropy when iterating a random discrete function

Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases} 1/m & \text{if } x\in S \\ 0 & \text{if } x\not\in…
fgrieu
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2 answers

Prof. Knuth lecture about $ \pi $ and random maps

In this video, Prof. Knuth talks about an interesting combinatorial problem: suppose you have a random map $ f\colon \{ 1, 2, 3,\ldots, n \} \rightarrow \{ 1, 2, 3,\ldots, n \}$. If you consider the values $( 1, f(1), f(f(1)), \ldots \}$, what is…
user967210
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6
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2 answers

Expected tail and head length of $\rho$ for a finite random function

Let $F: D \rightarrow D$ be a random function on finite domain $D$ of size $n$. It is well-known that, from any $x \in D$, iterating $F$ on $x$ traces out a sequence of values $x, F(x), F(F(x)), \ldots$ that must eventually repeat (since $D$ is…
Fixee
  • 11,760
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1 answer

First approximation of the expected value of the positive part of a random variable

Consider a random variable $X$ with mean zero ($\mu_X = 0$), known variance ($\sigma_X^2$), and all other moments finite but unknown. I am interested in obtaining an estimate of the expected value of the positive part of this random variable, i.e.,…
anon
5
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1 answer

Generate random numbers between a range such that no number comes twice.

Sorry if my question is stupid, math has been always a wild beast for me. I am an application developer. In one application I have a module which assigns a random 6-8 digit number and a serial number to a manager. The manager can then give these two…
Ratna
  • 155
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1 answer

Questions about generating non-biased random natural number

A. Several years before, I was solving some problems, and one of problems was something like Explain how you can get non-biased random natural numbers between 1~10, with a six-sided(normal) dice. I found the solution easily, but I was quite…
JiminP
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5
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2 answers

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 $O(n)$ time? Most sampling techniques I have in…
darksun
  • 341
5
votes
1 answer

Convergence of the fdds vs convergence in distribution in a function space

I'm trying to understand the essential difference between two common types of the convergence of random processes: the weak convergence of the finite-dimensional distributions (fdds) and the convergence in distribution in some function space (for…
Cm7F7Bb
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