Use this tag only if your question is 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 ...

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55 views

Exponential moments and integrability

Given a probability distribution $P(x)$ for which the moments $\int x^rdP(x)$ and the exponential moment $\int e^{Rx}dP(x)$ exist for some positive integers $r$ and $R$, than is it true that ...
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84 views

Representation of symmetric functions

Let $n \in \mathbb{N}$. Show that every symmetric function $f\colon E^n \rightarrow \mathbb{R}$ can be written in the form $f(x) = g\Bigl(\frac{1}{n}\sum_{i=1}^n \delta_{x_i} \Bigr)$, where $g$ has ...
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49 views

Why is the first equality of this proof valid?

Proposition: Given a sequence $(Y_n)_{n \in \mathbb{N}}$ of real-valued random variables on $(\Omega, \mathcal{F}.\mathbb{P})$, if $Y_n \to Y $ in probability and $\sup_{n \in \mathbb{N}} ...
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124 views

Strong Markov property of Bessel processes

I am thinking about the following: If $(B_t)_{t \geq 0}$ is a Brownian motion in $\mathbb{R}^3$, how can we show that the Bessel process (of order $3$) $(|B_t|)_{t \geq 0}$ has the strong Markov ...
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1answer
103 views

On non-existence of two independent random experiments on a same probability space

I have a question on Probability as follows: "Let $(\Omega,\mathcal{F},P)$ be a probability space. Then, on $\Omega$ we cannot construct two independent random experiments in which each experiment is ...
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32 views

Does Markov Chain converge in Variance Norm?

Assume the chain $\{X_n\}_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true ...
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30 views

Determinability

If $X$ and $Y$ are random variables taking values in measurable spaces $(E,\mathcal{E})$ and $(D,\mathcal{D})$ respectively, then we say that $X$ determines $Y$ if $Y=f\circ X$ for some measurable ...
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38 views

Find a specific probability distribution

Let $T \sim \mathcal{U}(0,1)$ and $X,Y$ such that $X=\Phi^{-1}(T)$ and \begin{gather*} Y = \begin{cases} -\Phi^{-1}(T+\frac{1}{2}) &\text{if}& 0 \leq T \leq \frac{1}{2} \\ ...
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1answer
48 views

$\lim\sup Z_n\leq \alpha , EZ_n\geq \alpha,|Z_n|\leq B \implies Z_n\rightarrow \alpha$ in probability

I come across a problem about converge in probability. The problem asks to prove the following: $$\lim\sup Z_n\leq \alpha , EZ_n\geq \alpha ,|Z_n|\leq B \implies Z_n\rightarrow \alpha\;\text{ in ...
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1answer
69 views

$\limsup$ and $\liminf$ of symmetric random walk

I have a question while studying the probability theory. Let $X_i$ be iid with $P(X_i=1)=P(X_i=-1)=1/2$. Put $S_n=\sum_{k=1}^nX_k$. Then how can I show that $\limsup_n S_n/\sqrt{n}=\infty$ and ...
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43 views

conditional porbability or not?

I have this exercise "Let X and Y independent random variables, with $$X \geq 0$$ e $$Y \sim exp (\lambda)$$ . Then prove that $$Pr(Y>X)=E(e^{-\lambda X})$$ " I have tried to use the conditional ...
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37 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
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1answer
85 views

Studying the probability of an event

Let $W:=(W_1,W_2,W_3,...,W_k)$ be a random vector of dimension $k \times 1$ taking values $(1,0,0,..,0)$, $(0,1,0,...,0)$,$(0,0,...,1)$. Assume that $W$ has a discrete uniform distribution. Let ...
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1answer
28 views

Finding measure given constant margins

Suppose $g:[0,1]^2\to R$ and $g$ can have finitely many discontinuities. $F$ is continuous and atomless c.d.f on $[0,1]$ $$\int_{[0,1]} g(x,y)dF(y)=1/2, \forall x$$ $$\int_{[0,1]} g(x,y)dF(x)=1/2, ...
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1answer
36 views

argument technique to prove convergence of random variable

I witness a lemma in my class note and I think the proof is not quite clear. Could anybody give me some ideas about argument technique to prove the lemma? The lemma 3 in the beginning of the text: ...
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46 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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1answer
43 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
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2answers
81 views

Probability with powers of $2$

There are $2^H$ tickets. Let $k<H$ (and so $2^k$ is a submultiple of $2^H$). Let $D$ be a random integer number taken uniformly from $1,2,..k$. Then $2^D$ is in turn a submultiple of $2^k$ and ...
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1answer
108 views

Change the order of conditional expectation of integration

I encountered this problem when learning SDE: $g(t,\omega)$ is a adapted process then $$\mathbb E\left(\int_a^b |g(t)|^2 \, dt \mid \mathcal F_a\right)=\int_a^b\mathbb E\left(|g(t)|^2\mid \mathcal ...
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42 views

Independence of increments of some processes

I am stuck on this question: Let $(B_t)$ be a standard Brownian motion. Define $$ (\tau_1)_t := \inf \{s \geq 0 : B_s = t \} ; \quad (\tau_2)_t := \inf \{s \geq 0 : B_s > t \}. $$ Any ideas how ...
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1answer
53 views

Joint density invariant under orthogonal transformations [duplicate]

I have a problem and I am totally stuck! I have to show that when the distribution of two random variables $X$ & $Y$ given by $g(x,y)=f(x)f(y)$ is invariant to orthogonal transformations, then it ...
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1answer
83 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
40 views

How to give a good upper bound on tail probability for $P\{|\frac{R_n}{\sqrt{n}}-1| \ge \varepsilon\}$?

Suppose $X_1,X_2,\ldots$ is a sequence of i.i.d. standard normal random variables. $R_n=\sqrt{X_1^2+\ldots+X_n^2)}$. How could I prove $P\{|\frac{R_n}{\sqrt{n}}-1| \ge ...
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37 views

Is set of product distributions compact under second moment constrains?

Please do not treat this question as duplicate of Definition of the set of independent r.v. with second moment contstraint which I didn't want to edit because of many useful comments. Also, in this ...
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1answer
41 views

Set of product distribution is the cross product of its sections

Let $(X_1,X_2)$ be an independent random pair with distribution $F(X_1,X_2)$. Let \begin{align*} S&=\left\{F(X_1,X_2)\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E[X_1^2] \le 1, \ E[X_2^2] \le 1 ...
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127 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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171 views

If $U$ is uniformly distributed on $S^{2}$, then its first component is uniformly distributed on $(-1,1)$.

Assume $\mathbf{U} = (U_{1}, U_{2}, U_{3})' \sim unif(S^{2})$. How would I show that $U_{1} \sim unif(-1,1)$? I don't know if I'm confusing myself, because I can't see this as being true for higher ...
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58 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty ...
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1answer
49 views

Independence and uncorrelatedness between two normal random vectors.

If $X$ and $Y$ are normal random vectors in $\mathbb R^n$ and in $\mathbb R^m$, and they are jointly normally distributed i.e. $(X,Y)$ is normally distributed in $\mathbb R^{n+m}$, then are the ...
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1answer
66 views

Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not generally true that $E\left(e^X\right)=e^{E(X)}$ (we can only use Jensen's Inequality here). ...
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34 views

Convergence of a distribution function

I have a problem that has me stumped. Frustratingly enough I can't really get the problem started. It goes as follows: Let $X_{1},X_{2},...$ be independent $C(0,1)$-distributed random variables. ...
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1answer
77 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
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68 views

Composition of random variable with its distribution is uniform

I'm trying to solve the following problem (Exercise 11.13) from Probability Essentials by Jacod and Protter: Let $X$ be a random variable (on $\mathbb{R}$) with distribution $F$ that is continuous. ...
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1answer
40 views

A question about random walk similar Markov Chain

This is an exercise from Probability and Measure by Billingsley: Let $(X_n)$ be a Markov chain with $\Bbb{Z}$ as its state space and with transition probabilities $(p_{ij})$ such that $$ ...
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1answer
211 views

Conditional expectation of the maximum of two independent uniform random variables given one of them

Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$. What is the conditional expectation of $\max\{X_1,X_2\}$ given $X_2$? And the conditional expectation of ...
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2answers
153 views

Expectation of Truncated Random Variables

Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $0<\delta<0.5$ and $\epsilon >0$ and define ...
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1answer
50 views

Finding $E(X \mid X > Y )$ when $X, Y \sim N(0,1)$ [duplicate]

This is the problem: The random variables $X$ and $Y$ are independent and $N(0,1)$-distributed. Determine $E(X \mid X > Y )$, $E(X + Y \mid X > Y )$. I go by the definition ...
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1answer
49 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
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1answer
53 views

Literature request on Markov Chains which state transition probability matrix evolves over time

I want to know is there any literature on markov chains who's state transition probability matrix evolves over time? For instance, I have 2 states, 1 and 2. With $$P = \begin{bmatrix} p_{11} & ...
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130 views

Proof about Inclusion-exclusion formula?

The problem requires to use the indicate function to prove the Inclusion-exclusion formula. But I really don't know what to do. Anyone can help with that? Thanks!
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1answer
55 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
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53 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
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1answer
75 views

Probability Question: Bridge problem

There are $n$ islands in the ocean. Each island is linked by a single bridge between each and every unique pair of islands to ensure no island is isolated from the others. The probability of each ...
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1answer
430 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
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1answer
46 views

Random variable bounded by another random variable

How to find $\Pr(z<X<Y)$ if $X$ and $Y$ are independent exponential r.v.'s with parameters $\lambda$ and $\mu$ So $x$ is bounded by $z$ and $y$, and y must be go from $z$, (and not from ...
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63 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
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31 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
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1answer
73 views

Two notions of total variation norms

I found these two definitions of the total variation norm for probability measures on $(X,\mathcal{F})$: $$ \left \|\mu- \nu \right \|_{TV} = \sup_{\text{$f:X \rightarrow [-1,1]$ measurable}} \left ...
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1answer
87 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
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1answer
46 views

Characterization of conditional independence

Definition: Let $\mathcal{G},\mathcal{K},\mathcal{H}$ be $\sigma $-subalgebras of $\mathcal{F}$, where $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ is a given probability space. We say that ...