Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
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Expected value of the number of draws with replacement to get all events at least once
Each time we draw, we get one of $N$ events $\{E_1,E_2,...,E_N\}$ with a fixed, but different probability for each event :$$P(E_1)=p_1\ne P(E_2)=p_2\ne P(E_3)=p_3\quad \text{etc.}$$
The events are ...
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L2 error of posterior mean - Infinite dimensional parameter set
I am working on a bayesian problem in function space. In particular I have consistency of the posterior measure and want to show that the posterior mean also adopts the convergence rated.
I have shown ...
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Sample complexity bounds of $L_S(h)$
Fix $\mathscr{H} \subset \mathscr{Y}^\mathscr{X}$ and a loss $\ell : \hat{Y} \times Y \to [0,1]$. Fix $S \in (\mathscr{X} \times \mathscr{Y})^{2m}$. Assume for now that $S$ is not random. Suppose we ...
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Which person out of the following 2 has more probability of winning, the one who goes first, or the second? How much is the probability of each?
Note: Before starting, I would like to point out that I have no concrete reason as to why I need to find this, but I just wanted to know.
Here is a bit of a background
There is a 2 player game where ...
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1
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What are the conditions on a probability distributions such that the expected time to encounter a nonzero event (or events) is finite?
I know that any specific event (or a specific subset of events) with a non-zero probability will occur almost surely (i.e, with probability 1) at some point given an infinite sequence of events.
My ...
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Probability of a 2-Player, Alternating Dart Game
I found an interesting probability question in my college textbook, but my solution differs from the provided answer. I hope this issue can be resolved here.
The Problem:
The following is the PMFs of ...
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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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Evaluate Conditional distribution and Conditional expection
This exercise is from J.F Le Gall GTM294, Measure theory, probability and stochastic processes.
Let $a,b\in(0,\infty)$, $(X,Y)$ be a random variable with values in $\mathbb{Z_+}\times \mathbb{R}_+$, ...
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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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Probability of an event involving the maximum of (stochastically ordered) random variables
Let $X_1,\dots,X_n\geq 0$ be independent, continuous random variables, and assume that $X_i$ stochastically dominates $X_j$ if $i<j$. This means that for any $x>0$,
\begin{align}
\mathbb{P}(X_i\...
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Calculating some Gaussian ratios
Let $N \geq 1$ be a positive integer, and let $w = (w_1, \dots, w_N)$ denote a positive sequence of real numbers. Let $\{g_n\}_{n \leq N}$ denote a sequence of iid standard Normal random variables.
...
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Distribution of gaps between observations in a specified window
I encountered something that looks like an interesting probability problem and was wondering if it (or something similar) has a known solution. Pointers on how to approach this would be greatly ...
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Why is it harder to prove lower bounds on expected maxima (than upper bounds)?
I am currently learning upper and lower bounds for the expected maxima of random variables, e.g. bounds for the quantity:
$$
\mathbb{E}\left[\max_{1 \leq i \leq n} X_i\right]
$$
Most of the ...
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Deterministic part of Lévy-Khinchin formula for measures on $[0,\infty)$
I’m working on an exercise involving the deterministic part of the Lévy-Khinchin formula for an infinitely divisible probability distribution on $[0,\infty)$.
A probability measure $\mu$ on $\mathbb R$...
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