# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

5,261 questions
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### Strictly Proper Scoring Rules and f-Divergences

Let $S$ be a scoring rule for probability functions. Define $EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$. Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...
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### A clean upper bound for the expectation of a function of a binomial random variable

I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
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### Probability and profit [on hold]

I have the following problem: I am taking trades. The probability that I win any given trade is 0.6. The probability that I lose any given trade is 0.4. If I win 3 times in a row, I win 7. If I win ...
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### Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors

Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
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### Skorokhod representation for weak convergence of exchangeable arrays

Let $(X_e)=(X_e)_{e\in \mathbb{N}^{(k)}}$ be a $k$-dimensional exchangeable real random array (see this note for the definition), where $k\in \{1,2,\ldots \}$ is fixed and $\mathbb{N}^{(k)}$ denotes ...
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### Expected size of matchings in a cubic graph

Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$? In other ...
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### Mixing time and spectral gap for a special stochastic matrix

Conisder the following dimension stochastic matrix, \begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &...
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### Theorems like the Lovász Local Lemma?

The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent. What other theorems exist in this genre? That is, what other theorems have ...
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### concentration inequality for a weighted sum of independent but not identical binary variables

Let $\alpha\in[0,1]$ be a fixed constant, and let $w,x\in[0,1]^n$ be two vectors such that $\sum_i w_i x_i=\alpha$. Define $Y = \sum_i w_i X_i$, where $X_i \sim \operatorname{Bernoulli}(x_i)$, so it ...
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### Monotonicity of $\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c \right]$ in $n$ for $c>\mathbb{E}(X).$

The nice question below was answered in the affirmative in On the sum of uniform independent random variables Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. ...
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### Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
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### How should the proof of the XYZ theorem be understood?

The XYZ Theorem of Shepp  states that for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for any three ...
Given a marked Poisson process in one dimension $$Y(t)=\sum_{\{t_i,a_i\}}g(t-t_i,a_i)$$ so that $Y(t)$ is a sum of impulses arriving as a Poisson process and the impulses $g$ belong to a family of ...