# 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 $n+1$ ...
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 ...
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### 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; ...
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### 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 ...
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### 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 $t \mapsto f(B_t)$ ?
1 vote
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### 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 ...
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• 105
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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 ...
• 183
1 vote
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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 ...
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 ...
1 vote
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, ...
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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 ...
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1 vote
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### 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 ...
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### 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 ...
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 ...
1 vote
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 ...
1 vote
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, ...
1 vote
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 $Y$ ...
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### 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. $$U_m \sim \mathcal{N}(\mu, \sigma)$$ ...
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### 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 ...
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### 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 ...
1 vote
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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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1 vote
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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. ...
1 vote
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 ...
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1 vote