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.
45,950 questions
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A theoretical problem seems related to conditional probability
Look picture,this is a problem seems related to conditional probability,The Q(t,A) seems like P(A|T=t),and P(A|T)=P(A|T=t) when T=t,this is the definition of conditional probability in elementary ...
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Expected Decay in the Number of Black Balls Over Iterative Sampling
Problem Statement
We begin with $N$ black balls. At each iteration, denoted by $\mathcal{D}$ steps in total, we perform the following operation: during the $i$-th step, we introduce $k^{i}N$ white ...
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Prove for value at risk $V_\alpha(-L)=-V_{1-\alpha}(L)$
Let $V_\alpha(L)$ be the value at risk of a function $L$ at $\alpha\in[0,1]$,
$$V_\alpha(L)=\inf\{x:P(L\le x)\ge\alpha\}.$$
Prove that for any $\alpha$,
$$V_\alpha(-L) =-\lim_{\epsilon\rightarrow0^+}...
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How to make a fair game of chance that discards further steps when a user inputted threshold is reached
I'm currently programming my own little game and I want to make it as fair as possible. If played over a long time one should neither lose nor gain money. I only had probability lessons back in high ...
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If $\sigma_1,\ldots,\sigma_n>0$ are given, can one find correlated $X_i\sim \mathcal N(0,\sigma_i^2)$ such that $X_1+\ldots+X_n=0$ almost surely?
Let $\sigma_1,\ldots,\sigma_n>0$ and $X_i\sim \mathcal N(0,\sigma_i^2)$, $i=1,\ldots,n$. Write $\sigma_{ij}=\mathrm{Cov}(X_i,X_j)$, $i,j=1,\ldots,n$, and $\Sigma=(\sigma_{ij})_{i,j}\in \Bbb R^{n\...
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A probability problem about gifts
In a party, $N$ people attend, each of them brings $k$ gifts. When they leave, each of them randomly picks $k$ gifts. Let $X$ be the total number of gifts which are taken back by their owners. Let's ...
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A probability problem about series [duplicate]
Problem: Let $ f(x) = \sum_{n=1}^{\infty} a_n x^n $, where $ \{a_n\}$ are independent random variables with $ P(a_n = 1) = P(a_n = -1) = \frac{1}{2} $. Show that $f(x)$ has infinitely many zeros in $[...
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Characterize the law of real random variable [closed]
Case 1: Suppose we have a probability space ($\Bbb{R}$, $B(\Bbb{R})$, $\delta_0$), if there is another probability measure $v$ satisfying $v(\{0\}) =1$. Can I say $v=\delta_0$?
Case 2: ($\Bbb{R}$, $B(\...
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Reframing Bertrand's chord problem as a line problem
Bertrand's famous question: "One draws at random a chord in a circle. What is the probability that it is smaller than the side of the inscribed equilateral triangle?" The consensus has long ...
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A Canonical-Measure Resolution of Bertrand's Chord Problem [closed]
I would appreciate feedback on this draft paper which purports to be a resolution of Bertrand's Paradox. Here are the Abstract, Summary, and Conclusion sections. I have posted the full draft to https:/...
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expected value of supremum of a exponential martingale [closed]
Consider the process
$$Z_t = e^{\int_0^t X_s dW_s - \frac{1}{2}\int_0^t X_s^2 ds}$$
where $X$ satisfies the Novikov's condition. What other hypotesis are required on $X$ to have
$$\mathbb{E}[\sup_{t \...
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Is the space of CDF on non-negative random variables compact in the space $[0,1]^{[0,\infty)}$ with product topology?
Let $F$ be the space of cumulative distribution functions on non-negative random variables which is the subspace of the space $M = \{f:f \text { is a mapping from } [0,\infty) \to [0,1]\}=[0,1]^{[0,\...
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Adaptive Betting on Roulette: Strong Law for Martingale Differences
Suppose that a player has at disposal two wagers at a roulette: bet one unit on a red or bet one unit on the single number $0$. It is well known that both wagers have the same house advantage, and ...
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Orbit of Brownian motion is dense a.s.?
I'd like to know what kind of stochastic processes in $\mathbb{R}^d (d\geq2)$ has dense orbit in $\mathbb{R}^d$ a.s.
For the first step, I'd like to check if an orbit of Brownian motion is dense in $\...
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Density of Ito Integral
Let $X_t$ be a d-dimensional process defined by
$$X_t=\eta+\int_{t_0}^t\sigma(s,X_s)dW_s,$$
where $\sigma(t,x)$ is a uniformly Lipschitz function w.r.t. x, and it is uniformly elliptic, namely, $\...
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