Questions tagged [probability-theory]
For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.
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Intuition behind the exponential convergence(e-convergence)
I'm studying a concept called e-convergence for sequences of probability densities. The definition states:
A sequence $(g_n)_{n \in \mathbb{N}}$ in $M_{\mu}$ is e-convergent to $g$ if:
$(g_n)_{n \in \...
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Polya urn scheme by induction task
I've been trying to solve the task given in "An Introduction to Probability Theory and Its Applications" by William Feller. The task is to show by induction that the probability of a black ...
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Confusion on defining uniform distribution on hypersphere and its sampling problem
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ...
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Random variable support definition
Looking at the definition of support of a random discrete variable I have come upon two different definition of support:
The first one is defined by the set $$\{x:P(X=x)>0\}$$
The second one ...
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Probability of two independent binomial random variables are equal
Let $X_{1},X_2\sim Bin(n,p)$ be iid binomial random variables. Then it is intuitive that $\mathbb P(X_1=X_2)$ decreases when $n$ increases (keeping $n$ fixes). However, it is not obvious to prove this ...
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There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$.
By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution.
I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
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Product of 2 normal variables with positive means
Suppose $X \sim N(\mu,1)$, $Y \sim N(\mu,1)$ are iid normal random variables with $\mu>0$. My research problem is finding out the asymptotics of the tail function of XY (since the explicit formula ...
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conditional probability with respect to the stopped sigma algebra
Suppose that $\sigma$ is a almost surely finite stopping time with respect to some filtration $(\mathcal{F}_t)_{t\in\mathbb{R}}$, and let $X$ be a real walued random variable defined on the same ...
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Probability of picking opposite sides of a unit square
The unit square is the square spanned by the points $(0,0),(0,1),(1,0)$, and $(1, 1)$ in the plane. Two points are chosen uniformly on the perimeter of the unit square. Find the probability that the ...
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Definition of a Markov process
I found 2 Definitions for a Markov process and I am trying to understand how they are connected.
Let $X=\left(X_t\right)_{t\geq 0}$ be a $\mathcal{F}_t$ adapted Process. We say $X$ is a Markov ...
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Intuition for bounds of Adaptive Conformal Inference
I have been reading the paper by E. Candès and Gibbs about Adaptive Conformal Inference (here is the original papel). The main idea is to update the miscoverage level $\alpha_t$ as
$
\begin{cases}
\...
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LLN's that can be applied when the size of the vector also goes to infinity?
suppose $\hat{\mathbf{X}}$ is an an empirical average of $\mathbf{X}=\left[\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\right] \in \mathbb{R}^{p \times n}$
I had written the following only to be ...
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If $X_n$ is martingale, $N$ is a stopping time, is $X_{n+N}$ a martingale?
Is this true? If it is, can we change martingale to sub or super?
My attempt (On submartingale): $\mathbb{E}[X_{n+N+1}\vert X_{n+N}]=\mathbb{E}[\mathbb{E}[X_{n+N+1}\vert N,X_{n+N}]\vert X_{n+N}]\ge \...
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Questions in proving $\mathbb{P}\left(T_a<\infty\right)=1$ with $T_a:=\inf \{t>0: B_t \ge a\}$
Let $\left(B_t, t \geq 0\right)$ be a one-dimensional Brownian motion starting from the origin (i.e, $\left.B_0=0\right)$. Let $\mathcal{F}_t:=\sigma\left(B_s: s \leq t\right)$ be the filtration ...
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The Riesz representation theorem and probability density functions?
The Riesz representation theorem asserts that if the linear functional $L:C[a,b]\rightarrow\mathbb{R}$ is bounded (and hence continuous), then there exists an $\alpha\in BV[a,b]$ with $\operatorname{...
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