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.
248 questions
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Calculating the probability distribution of steps a random affine recurrence/walk takes to reach atleast a certain value
Assuming i have $n$ affine functions (of the form $f_i := x \mapsto a_i x + b_i$ where $a_i, b_i \in \mathbb{R}$). I have a random, discerte process andapply them like this $y_{i+1} = f_r(y_i)$ where $...
- 351
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Determining the distribution of Gaussian Process at random time
There is a rather intuitive result in Gaussian Processes, it goes as follows:
Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a sample-continuous Gaussian Process. Let $T_1, T_2: \mathbb{R} \rightarrow \...
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Hessians of Gaussian Random Fields at Minima and Stationary Points
I'm interested in carefully quantifying the number of local minima and stationary points of a Gaussian random field. While working through it, I came across a puzzling conclusion that doesn't seem ...
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Matrix-Free Stochastic Approximation
Let $C: \mathbb{C}^m \to \mathbb{C}^m$ be a sparse Hermitian operator that I only know via matrix-vector products. Using Chebyshev filtering, I obtain the k highest eigenvalues $\lambda$ and ...
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Regular conditional probability distribution of a stochastic process
Let $X=(X_t)_{t\in T}$ be a stochastic process, perhaps in a polish space $(E, \mathcal E)$. Under what conditions does a regular conditional distribution for $X$ exist? I.e.
when does a regular ...
- 3,089
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Equation in distribution [closed]
Are there examples of probability distributions (non deterministic) $\mu_x, \mu_y$ on the positive reals $(0, \infty)$ such that for two independent random variables $X \sim \mu_x$, $Y\sim\mu_y$ ...
- 642
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Expectation of a bivariate function vs the expectation of the transformed variables
Let $(X_1,X_2)\sim f(x_1,x_2)$, also $\phi=g(x_1,x_2)$, where $g$ is invertible with respect to both $x_1$ and $x_2$, we can obtain the density of $\phi\sim p(\phi)$ through the Jacobian rule.
My ...
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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 ...
- 1,518
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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 $n+1$ ...
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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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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 ...
- 699
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102
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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 ...
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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 $c$. ...
- 651
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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 ...
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