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.
44,511
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Large deviation principle continues to hold when rate function is replaced by its lower semicontinuous regularization
Consider
Lemma
Suppose $I$ is a function such that (2.1) holds for all measurable sets A. Then (2.1) continues to hold if $I$ is replaced by $I_{\text{lsc}}$
proof:
$I_{lsc}\le I$ and the upper bound ...
- 3,120
1
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1
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102
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Intuition behind measurability of Markov kernels
Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces. In the definition of Markov kernel $K:X\times\mathcal{B}\rightarrow[0,1]$, it is required that
$K(x,\cdot)$ is a probability measure ...
- 113
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General proof tactic for showing a.s. limit equalities?
I am interested in problems of the form:
Show that $$\lim_{n\to \infty} X_n=C \text{ a.s.} $$
where $C$ is some known constant.
I would like to know if there are any "classic" ways of ...
4
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1
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167
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Show a convergence in probability using Markov inequality
Let $(X)_{j\in \mathbb Z}$ be a discrete random process such that
$$X_{j}= \theta X_{j-1}+ \epsilon_j, \quad (\epsilon_j) \overset{iid}{\sim} N(0,1), |\theta|<1 $$
How to show that
$$P\left(\max_{1\...
- 351
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1
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46
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Total Correlation is difference of relative entropies in general?
Motivation
Consider a finite set $[q]=\{1,\dots,q\}$, random variables $X_1,\dots,X_k\in[q]$, and their product $X=X_1\otimes\cdots\otimes X_k\in[q]^k$, i.e. the components of $X$ are independent.
Let ...
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Probability Notations and Continuity
I started working on probability theory and there is something I am confused with.
Consider a transition function p on a countable and compact space $X$ that is $p(x^\prime|x)$.
How can I understand ...
- 3
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78
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Double Summation over a Subset of a Cartesian Product
From the "Probability & Statistical Inference, 9th edition" by Hogg, Tannis, Zimmerman, it is stated that one of the properties of the Joint Probability Mass Function of Random Variables ...
- 93
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A simple example to practice justifying the interchange of derivative and expectation
I have been self-studying probability and measure theory. The condition I have for the interchange of derivative and integral is as follows, and comes from Folland, G. B. (1999). Real analysis: ...
- 952
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Prove posterior mean is positive if and only if signal is positive, assuming zero prior mean [closed]
Suppose that:
$\Delta \sim G$, where $G$ is a distribution that is symmetric about the origin
I get a normally-distributed signal:
$\hat{\delta} \: |\Delta, \tau^2 \sim N(\Delta, \tau^2)$
How can I ...
- 29
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99
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Symmetric $\alpha$-stable distributions
Consider the expression appearing in the Levy-Kchinchine formula:
$$\varphi(t) = e^{ita - \sigma^2/(2t)+\int_{-\infty}^\infty [e^{itx} - 1 -itx \mathbf{1}_{|x|<1}]\,{\rm d}\nu(x)},$$ but with $a =0,...
- 281
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Limit of an integral related to characteristic functions
Suppose $X$ is a random variable with CDF $F_X$ and characteristic function $\varphi_X(t) = Ee^{itX}$. The inversion formula for the characteristic function gives:
$$
F_X(b)-F_X(a)=\frac{1}{2 \pi} \...
- 2,093
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Reducing integrals over abstract spaces to integrals on $\mathbb{R}_{+}$ with respect to the Lebesgue measure
I am studying for a measure-theoretic based course in probability theory and I am stuck on a theorem that goes as follows.
Definition
First, it defines a measurable space $(E,\mathcal{E})$ to be ...
- 373
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How can we rigorously show conditional independence here?
Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda\in(0,\infty)$$ and $\mu$ denote the measure with density $\frac pc$ ...
- 13.4k
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Categorical probability theory: Fritz' diagram categories
1. Context.
In the seventh chapter of his 2019 paper A synthetic approach to categorical probability theory Tobias Fritz introduces certain diagram categories. Supposedly, these allow for a ...
- 3,018
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1
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Proving function of a martingale is integrable and measurable [closed]
Let $(X_n)_{n\in \mathbb{N_0}}$ be a martingale with values in $\mathbb{Z}$ such that $|X_n - X_{n-1}| = 1$ and $f: \mathbb{Z} \rightarrow \mathbb{R}$ an arbitrary map. I need to show that $f$ is ...
- 876
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1
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Proving Hoeffeding's lemma [closed]
For $X$ a real-valued random variable such that $|X| \leq 1$ a.s., I need to prove the following by using the Jensen's inequality:$$\mathbb{E}[X] = 0 \implies \mathbb{E}[e^{\lambda x}] \leq \cosh(\...
- 876
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1
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46
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Prove the existance of a random variable satisfying a given property [closed]
We are given a real-valued rv $X$ with $|X|\leq 1$ a.s. I need to show that there exists a rv $Y$ with values in $\{-1,+1\}$ such that $\mathbb{E} [Y|\sigma(X)] = X.$ I do not know how the conditional ...
- 876
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1
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Generalization error of a simple perceptron for a student-teacher network
I have been asked to prove the following expression given the following density probability function for a student-teacher perceptron network
\begin{equation}
P(x,y) = \frac{1}{2\pi\sqrt{Q-R^2}} \cdot ...
- 13
3
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2
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35
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Show that the probability: $P(A^c \cup C) \geq 7/8$
Let $A,B,C$ be pairwise independent events with $P(A \cap B) = 0.1$ and $ P(B\cap C)=0.2$. Show that $P(A^c \cup C) \geq 7/8$.
This is what I could do
Since $P(B) \geq 2/10$ and $P(A)P(B)=1/10$, $P(A) ...
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59
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Arrival times of trains
Let $N^O$ and $N^U$ be an independent pair Poisson processes with intensities $\alpha_O$ and $\alpha_U$. We will model the incoming trains in the central station (trains that are arriving above the ...
- 35
4
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1
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81
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Law of iterated logarithm and hitting time distribution
Let $X_i$ be a collection of independent standard normal variables and $S_n = \sum_1^n X_i$. The by the law of iterated logarithm
$${S_n \over \sqrt{2n \log \log n}} > 0.8$$
infinitely often, where ...
- 5,538
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0
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30
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log-likelihood: changing relative contribution of two summation terms
Suppose I have a log-likelihood of the form
$$\mathcal{L} = \sum_{i = 1}^{n} a_i + \sum_{j = 1}^{m} b_j,$$
where $a_i$ and $b_j$ are some independent log-probabilities. The problem is, the second sum $...
1
vote
2
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137
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Calculate $E[X|X+Y=z]$ for independent and identically distributed $X$ and $Y$.
Let $X,Y$ be independent and indentically distributed random variables taking values in $(0,\infty)$ and $z>0$. I want to calculate $E[X\mid X+Y=z]$. My intuition tells me, that its value should be ...
0
votes
1
answer
120
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Expected value of the reciprocal of sample standard deviation
Hopefully I am posting this in the right place, I've never used StackExchange before.
The question I am answering is:
Assume that X1, . . . , Xn are iid from $N(µ, σ^2)$ with unknown µ and $σ^2$.
Find ...
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35
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Ash Probability and Measure 1.2.6 Proof Check
I was hoping someone would check my proof. The question is Let $f: \Omega \to \Omega'$ and let C be a class of subsets of $\Omega'$ Show that $\sigma(f^{-1}(C))=f^{-1}(\sigma(C))$
My proof is below.
...
3
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1
answer
96
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The Sum of Independent Poisson Random Variables is Poisson: Converse
Suppose we have random variables $X \sim \text{Poisson}(\lambda)$ and $Y \sim \text{Poisson}(\mu)$ such that $X+ Y \sim \text{Poisson}(\lambda + \mu)$. Must $X$ and $Y$ be independent?
Contrasting ...
- 1,321
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The optimal constant for Burkholder-Davis-Gundy inequality
I have searched for some posts on math stack exchange, but I do not find a clear answer, can I have an explicit optimal constant for the BDG inequality $$\mathbb{E}\left[\int^T_0 \big|X(s)\big|^2ds\...
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Boundedness of $\dfrac{W_2(\mu_1+\varepsilon (\mu_2-\mu_1),\mu_1)}{\varepsilon}$ for 2 -Wasserstein metric
Let $\mathcal{P}_2(\mathbb{R}^{n})$ the space of Borel probability measures of finite second moment in $\mathbb{R}^{n}$ equipped with the $2$-Wasserstein metric $W_2$. Let $\mu_1$, $\mu_2 \in \mathcal{...
- 445
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1
answer
115
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Existence proof of conditional expectation
I am self-learning introductory stochastic calculus from the text A first course in Stochastic Calculus, by Louis Pierre Arguin. I'm struggling to understand a particular step in the proof, and I ...
- 5,420
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1
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96
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Understanding the derivation of the Poisson process. [closed]
I am learning about the Poisson Process and at the beginning of the proof, the following statement is left as an exercise for the reader:
Let $N_{(a,b]}$ denote the number of random events in the ...
- 341
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1
answer
99
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Simulating a continuous-time jump process
I've got a continuous-jump process $(X_t)_{t\ge0}$ with generator $$(Af)(x)=\tilde\lambda(x)((\tilde\kappa f)(x)-f(x)),$$ where $$\tilde\lambda=\lambda+c$$ for some bounded measurable nonnegative $\...
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1
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180
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Choose a random number btwn 0 and 1. Keep doing it until the sum of the numbers exceeds 1. Expected value of this sum?
This is a modified version to a pretty common problem; here's that classic problem: Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $...
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Time Series Analysis and Recurrence Relations/Differential Equations
I am beginning to watch a video playlist on the subject of time series analysis, and it seems pretty clear both from notation and some of the terminology (such as "characteristic equation") ...
- 1,904
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1
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$\lim\limits_{n \to \infty}\frac{1}{n^2}\sum_{i=1}^n \mathrm{Var}[X_i] = 0$. Then $\frac{1}{n}\sum_{i=1}^n (X_i - \mathsf{E} [X_i])\to0$.
Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $(X_k)_k$ be a sequence of independent random variable with finite second moment. If $\lim\limits_{n \to \infty}\frac{1}{n^2}\sum_{i = 1}^n \...
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Show $\bigcup\limits_{n\mid n\geq N}A_{n,m}=\left\{\sup\limits_{n\mid n\geq N}\left|\frac{S_n(\omega)}{n}-p\right|\geq\frac{1}{m}\right\}$
Let be $m,n\in\mathbb{N}$, $p\in(0,1)$, $S_n$ is a binomially distributed random variable and
$
\begin{align*}
&A_{n,m}:=\left\{\omega\in\Omega\mid \left|\frac{S_n(\omega)}{n}-p\right|\geq \frac{1}...
- 4,493
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Dirac delta distribution and supremum
I have asked the same question here.
My doubt follows from this paper pg. 546. The context is they find the lagrangian function and the primal variable is the set of all density functions.
To make ...
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Does "$n$ related strategies such that at most one fails" imply a single strategy that fails with probability $\le 1/n$?
In a generalized "prisoners/hats/boxes"-style logic puzzle, suppose we have the following:
a set $W$ (of "possible world configurations")
a set $S$ (of "possible individual ...
- 11.5k
1
vote
0
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44
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Gauss-Markov characterising property
I am trying to solve exercise no. 1.13 page 86 of 'Continuous time Martingales & Brownian Motion', by Revouz-Yor.
It says that a stochastic process $X_t$ is a Guassian process (i.e. the marginal ...
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1
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77
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Show that $\frac{1}{n} \sum \limits_{k=1}^{n} X_{k}-\frac{1}{n} \sum \limits_{k=1}^{n} \mathbb{E} X_{k} \stackrel{\text { f.s. }}{\longrightarrow} 0$
Let $ \left(X_{n}\right)_{n \in \mathbb{N}} \subset \mathcal{L}^{2} $ be a sequence of independent random variables. Furthermore, let $ \alpha<1 $ and $ c \geq 0 $ be such that $ \sum \limits_{i=1}^...
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2
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equality of conditional expectation
Let $X_1,...,X_k$ be independent exponentially distributed with parameter $\lambda$ random variables.
Is this equality holds?
$$E[e^{-X_1-...-X_k}|X_1+...+X_k]=e^{-X_1-...-X_k}$$
My textbook says it ...
0
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1
answer
181
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100 People Throwing D100
If one person throws 100 times a fair dice with 100 corners (numbered from 1 to 100), then their probability to roll 100 at least once in any of those times equals:
$$P(X=100)=1-\left(\frac{99}{100}\...
- 273
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1
answer
81
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Show that $ \frac{Y_{1}+\cdots+Y_{n}}{n} \stackrel{\mathrm{P}}{\longrightarrow} \mu \text { as } n \rightarrow \infty $
Let $Y_{1}, Y_{2}, \ldots$ be a sequence of identically distributed random variables with $\mathbb{E} Y_{1}=\mu$ and $\operatorname{Var} Y_{1}=\sigma^{2}<\infty$. Furthermore, let $Y_{j}$ be ...
- 626
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0
answers
66
views
Essential Supremum over a set of random variables attaining maximum
I think the following result is true, but I cannot find any source online, so I post it here for confirmation:
Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$, an arbitrary index set $\...
0
votes
1
answer
115
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residual waiting time and spent time
Does the expected value of the residual time (the amount of time one has to wait) equal the expected value of the spent time (the amount of time since the last arrival)? I know that the expected value ...
- 85
0
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0
answers
33
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Existence of a Unif(0,1) random variable for counter-monotonicity
We know that for a pair of counter-monotonic random variables $X$ and $Y$, it holds that
\begin{equation}
(X,Y)\sim\big(q_X(U),q_Y(1-U)\big)
\end{equation}
for $U\sim\text{Unif}(0,1)$ and $q_X,q_Y$ ...
- 25
1
vote
0
answers
62
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Question about showing stopping time
I'm having a trouble about how to show $T$ is a stopping time. For instance, suppose $X$ is a simple random walk on $I = \mathbb{Z}/k\mathbb{Z}$, and $T$ is the first time $n$ such that $X_0, \dots, ...
- 21
2
votes
2
answers
87
views
Is the normal distribution where the parameters themselves are normal random variables still normal?
We denote by $\mathcal{N}(\mu, \sigma^2)$ the normal distribution where $\mu\in\mathbb{R}$ is the mean and $\sigma\in\mathbb{R}$ is the scale parameter.
Assume that the parameters $\mu,\sigma$ are ...
- 4,814
1
vote
1
answer
98
views
random walk, expected value of visiting point before back to origin
Consider standard random walk on integer line. (probabilities $\frac{1}{2}$ to go left or right, start at $0$).
I am reading probability and random processes book and I came across following theorem:
...
1
vote
0
answers
62
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derivation of exponential distribution's expectation using poisson
I know that for a Pois$(\lambda)$ distribution, the expectation is $\lambda$. I also know its relation to the exponential distribution, and how you can derive the PMF of the exponential through the ...
- 85
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1
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51
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reccurence relation, particular solution does not work
I have linear nonhomogeneous reccurence equation:
$$pa_n-a_{n=1}+(1-p)a_{n-2}+1=0$$
I solve homogeneous equation first and I get: $a_n=A+B(\frac{1-p}{p})^n$
Now I want to guess particular solution, so ...