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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13 views

To show $X$ and $|X|$ are not jointly continuous

Suppose $X\in N(0,1)$. Show that $X$ and $|X|$ are not jointly continuous. I am not sure how I can approach this problem. But the following method seems plausible to me: $$P(X\leq ...
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1answer
19 views

Proving that the mean of a random variable is continuous, where is dominated convergence being used?

I am looking at the proof of the first part of this lemma. Previously in the text another theorem was stated: Convergence in distribution, $Y_n \implies Y$, holds iff $Ef(Y_n) \rightarrow Ef(Y)$ ...
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1answer
9 views

A question on measurability of stochastic process

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{X_t:t\geq0\}$ be a collection of real-valued random variables with index set $[0,\infty)$. Show that the mapping $t\mapsto X_t$ is ...
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1answer
16 views

conditional distribution for a discrete RV

For a discrete RV $X$, is it true that the conditional distribution $P_{X \mid Y} (B \mid y)$ is discrete as well for all $y$? I only managed to prove that this is true almost surely. Let $\Pr(X\in ...
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1answer
33 views

Amazing property of martingales

let $Y_1,Y_2,..$ be a sequence of equally distributed, independent and positive random variables. Consider $X_n = Y_1…Y_n$. Under which condition is $X_n$ a (super)-martingale? Show that neglecting ...
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1answer
14 views

Evaluating limit of a characteristic function (Fourier Transform) in $R^k$

I am trying to evaluate this limit: $$\lim_{n \to \infty} \left[(\text{det}\ \Gamma_n)^{-\frac{1}{2}}\exp \left\{ {-\frac{1}{2}(x-m_n)\cdot (\Gamma_n)^{-1}(x-m_n)} \right\} \right]$$ where ...
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10 views

find a sequence of independent, non-identical r.vs , such that $limsup (X_1 + … + X_n) / n = \infty $ [on hold]

Find a sequence of independent, non-identical, non-negative random variables with $E X_i = 1$, such that, $$limsup_{n\rightarrow \infty} \dfrac{(X_1 + \dots +X_n)}{n} = \infty \ a.s.$$
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0answers
24 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
3
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1answer
17 views

Chromatic number $\chi(G)=600$, $P(\chi(G|_S)\leq 200) \leq 2^{-10}$

I am learning martingale and Hoeffding-Azuma inequality recently but do not how to apply the those inequality or theorem here. Let $G=(V,E)$ be a graph with chromatic number 600,i.e. $\chi(G)=600$. ...
4
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1answer
69 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
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0answers
12 views

Prerequisites to reading *Convergence of Probability Measures* by Patrick Billingsley.

I want to improve myself in asymptotic theory regarding the realm of probability. I tried reading Convergence of Probability Measures by Patrick Billingsley but right off the bat the De ...
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1answer
21 views

Probability of two RVs being equal

Let $X$~Binom($n,1/2$) and $Y$~Binom($m,1/2$) be independent. Calculate $P(X=Y)$. My attempt: Assume $m\le n$ $$P(X=Y)=\sum_{k=0,\ldots,m} P(X=k)P(Y=k)=(\frac{1}{2})^{n+m} \sum_{k=0,\ldots,m}{n ...
2
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1answer
30 views

Expected number of trails to get $n$ heads in a row with an increasing biased coin.

Assume that we have a biased coin with probability $p_1$ of getting H and $1−p_1$ of getting T on the first trial, $p_2$ of getting H and $1−p_2$ of getting T on the second trial and so on such that ...
2
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0answers
15 views

Which matrices are covariances matrices?

Let $V$ be a matrix. What conditions should we require so that we can find a random vector $X = (X_1, \dots, X_n)$ so that $V = Var(X)$? Of course necessary conditions are: All the elements on ...
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0answers
14 views

Probability Urn with N Coins [on hold]

An urn contains N coins among which Θ of them are perfectly balanced, (the probabilities for heads and tails is 0.5), while the remaining N - Θ coins are two headed. A coin is drawn at random, it is ...
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2answers
31 views

Question about Strong law of large numbers

I am confused by this problem. Let $F$ be a distribution function with $F(0-)=0$ and $F(1)=1$ and let $\mu$ be the associated law. Let $m_k=\int_{[0,1]}x^k dF(x)$. Define \begin{array}{c c c} ...
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1answer
15 views

Topology of weak convergence, linear functionals and probabilistic intuition

One very basic question regarding the topology of weak convergence. We know that given the following: $X$ metrizable topological space, $\mathcal{B} (X)$ Borel $\sigma$-algebra, $\Delta (X)$ ...
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1answer
44 views

Sum of i.i.d. random variables is a markov chain

I think I have some problem understanding markov chains, because we defined them as abstract objects but our professor does proofs with them as if they where just elementary conditional probabilities. ...
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2answers
29 views

What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean?

This is a bit of a silly question, but I can't seem to find the answer anywhere. I feel like $X\sim \mathcal{N}(\mu,\sigma^{2})$ means that $\sim$ is a relation, but if it is a relation, what ...
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1answer
22 views

Does the random variable $f(\tau)M_\tau$, where $M$ is a martingale and $\tau$ is a stopping time, have zero expectation?

Suppose that $M:=\{M_t\}_{t\geq0}$ is a martingale adapted to some filtration $\mathcal{F}:=\{\mathcal{F}_t\}_{t\geq0}$ with $M_0\equiv0$ and that $\tau$ is an $\mathcal{F}_t$-stopping time. Suppose ...
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0answers
10 views

implication of positive speed of random walk on a graph

Let $(V,E)$ be a vertex-transitive graph and let one vertex be the origin. Let $d(v,0)$ be the graph distance between $v$ and $0$. Consider $(X_n)$ a simple random walk on the graph. Let $A_n$ be the ...
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0answers
11 views

Interpretation of $\sigma$-algebra and filtrations (follow-up question)

This is a follow-up question to Interpretation of sigma algebra, particularly to Jun Deng's excellent answer. He used the example of two coin tosses to explain some fundamentals of how filtrations and ...
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0answers
15 views

Dependent events and the union bound

Observation Suppose we roll three fair dices, and obtain the numbers $n$, $m_1$, $m_2 \in \{1,2,3,4,5,6\}$. From simple counting, we know that $ \Pr_{n,m_1} [n < m_1] = \frac{15}{36} $, and ...
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2answers
69 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
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0answers
5 views

Factorization Theorem for Sufficient Statistic

To show that $S(X)$ is a sufficient statistic for $\theta$ I used the joint density of $X_1,...,X_n$: $$f_\theta(\mathbf x)=\mathbf1_{[\theta \le min(x_i)]}\mathbf1_{[max(x_i) \le \theta +1]}$$ But ...
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1answer
12 views

Bound Involving Submartingales

Let $(X_{j})_{j \geq 1}$ be a sequence of random variables with $X_{j}$ having mean zero and a finite moment generating function $\phi_{j}(\xi) = E(e^{\xi X_{j}})$ for all $\xi$ in a neighborhood $J$ ...
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0answers
16 views

Bayes classifier error [migrated]

I've recently been working my way through Elements of Statistical Learning and have been stuck on Exercise 7.2 for the last couple of weeks. The question states: For a 0-1 loss with $Y \in \{0,1\}$ ...
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1answer
22 views

Finding probability, piror probability, posterior probability [on hold]

Anyone help me to solve this: A survey organization randomly selects an adult Spainsh for a survey about credit card usage. Use subjective probabilities to estimate the following. 1) What is the ...
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2answers
47 views

If $X$ is standard Normal then find $\lim_{x\to0}P(X>x+\frac{a}{x}|X>x)$

If $X$ is Standard Normal and $a>0$ is a constant then find $\lim_{x\to0}P\big(X>x+\dfrac{a}{x}\big|X>x\big)$. This is an exercise from a book whose name I cannot immediately recall. I ...
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0answers
14 views

limit of characteristic function Normal

I need to find the limit of the following characteristic function as $s \rightarrow\infty$ $\frac{e^{-it\frac{s}{\sqrt{s^2+s}}}}{(1-(e^{-it\frac{1}{\sqrt{s^2+s}}}-1)s)}$ The top part seems to reduce ...
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1answer
32 views

What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
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0answers
15 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
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2answers
26 views

Solving the probability of independent evnts without the complement

Suppose that virus transmision in 500 acts of intercourse are mutually independent events and that the probability of transmission in any one act is $\frac{1}{500}$. What is the probability of ...
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1answer
13 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
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1answer
15 views

Independence of sigma algebras of sigma algebras

I have a bunch of questions all of which more or less fall under the subject in the title. The first one goes as follows. Let $E_1,E_2,\ldots,E_n$ be collections of measurable sets on ...
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1answer
18 views

Can an indicator function be a valid Radon Nikodym derivative?

Take a process $X_t$ defined on a canonical space with probability $\mathbb{P}$. Can the indicator function $\mathbb{1}_{X_t< U}$ be a Radon Nikodym derivative? That is can we have a measure ...
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0answers
9 views

Speed of convergence of squares of RVs

My problem appears to be pretty simple. I've obtained the speed of convergence — Berry-Esseen bound — for the expression $$ \underset{C \in \mathcal{C}}{\mathrm{sup}} | Q_{X, n} (C) - \Phi (C)| ...
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0answers
27 views

how to prove this function is a probability measure in $U_B$

Let $(\Omega, U, P)$ be a probability space. and $B\in U$, $P(B)\gt 0$ $U_B =\{A: A=B\cap C, C\in U\}$ its class in $\Omega$ is a $\sigma$-algebra and $P_B : U_B \to \Bbb R$ $A \to ...
2
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4answers
76 views

Must be Bayes' theorem

A professor gave me the following question a week earlier that he himself was doubtful about. I gave it quite a lot of thought but couldn't come up with any way to solve the question. Here it is:- ...
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0answers
17 views

Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:- A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
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0answers
15 views

Proving the desintegration theorem

Theorem: For a probability space $(\Omega,A, \mathbb{P})$, a standard Borel space $(S, B)$ and a measurable map $X: \Omega \to S$ there exists a markov kernel $k$ s.t. $$\forall C \in B: P(X \in C | ...
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1answer
58 views

Conditional expectation, $X = \varphi (Y)$

Show that if $$\forall \omega \in A \ : \ X(\omega) = \varphi(Y(\omega)), \ \ A \in \Sigma_Y$$ (that is, the equality is true for $\omega \in A$), then $$\mathbb{E}(X|Y)(\omega) = \varphi(Y(\omega)) ...
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0answers
12 views

Ross Intro to Probability Models--Example 4.4

Can someone explain to me please how we derived the Transition Matrix? Why we decided to put $P_{00} =0.7$ and $1 - P_{00} = P_{02}$. I just don't see it the way Ross defined the different states. ...
3
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0answers
20 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
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1answer
41 views

almost sure convergence for non-measurable functions

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume for each $n$, $Y_n:\Omega\rightarrow\mathbb{R}$ is a function but $Y_n$ is not necessarily $\mathscr{F}$-measurable. In this case, is it ...
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0answers
20 views

Do the eigenvectors of a random orthogonal matrix have Haar measure?

For orthogonal $Q$ with Haar measure, does the group of unitary matrices $U$ which diagonalize $Q=U\Lambda U^H$ have Haar measure? I'd be happy to know any answer, even if it's only true for certain ...
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0answers
9 views

Concentration of measure of inner product in Hilbert space?

In the finite dimensional Hilbert space of quantum mechanics (one where all vectors have norm one), is a concentration of measure phenomenon observed with the inner product of any two vectors? That ...
3
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0answers
56 views

Hitting time for a random walk on the grid

Consider the grid on $n^2$ nodes composed of points $x$ and $y$ with coordinates in the set $\{1, \ldots, n\}$, and consider the discrete-time Markov chain which transitions to a random neighbor at ...
1
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1answer
36 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
0
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0answers
24 views

Characteristic function of a product of two dependent random variables

If you're given the characteristic function of a continuous random variable, say $X$, and the distribution of another discreet random variable, say $U$, which is dependent of $X$, how do you ...