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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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
7 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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Bayes classifier error

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
16 views

Finding probability, piror probability, posterior probability

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
40 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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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
23 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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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
24 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
11 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
17 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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6 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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25 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 ...
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4answers
72 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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16 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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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
42 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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11 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. ...
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18 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
38 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
15 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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8 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 ...
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46 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 ...
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1answer
33 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 ...
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21 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 ...
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1answer
27 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
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26 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
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1answer
23 views

Random Variable Rescaling (Lindeberg CLT)

Suppose that $\{X_m\}_{m=1}^\infty$... is a sequence of independent random variables respectively distributed as $P(X_m = 1) = P(X_m = -1) = p_m$, $P(X_m = 0) = 1 - 2p_m$, $m \ge 1$, where ...
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26 views

What does $\mathbb P(\overline{\mathbf X} = \mathbf x)$ mean

I am reading Peter Hall's "the bootstrap and edgeworth expansion". In Theorem 2.3 on page 57, it claims that if the characteristic function $\chi$ of a $d$-dimension random variable $\mathbf X$ ...
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1answer
43 views

Moments of a random variable in terms of its cumulative distribution function

Consider a random variable $X$ with distribution function $F(x)$. Calculate the $r$th moment of $X$, $\mathbb E X^r$. I read that the desired moment can be calculated as follows. $$ \begin{align} ...
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12 views

A question about the translation property Markov kernel

Given that ${X_n}$ is a Markov chain, and a Markov kernel with translation propert$p(y+x,E+x)=p(y,E)$. Question:How to show $Y_n=X_n-X_{n-1}$ are i.i.d? I'm trying to use Markov Property and ...
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Why do we need ϵ>0 for typical set in Asymptotic Equipartition Property (AEP)?

In following text, author has used ϵ>0 for a typical set in AEP but it don't matter if we don't take it. Why is ϵ needed as I am seeing it a lot in information theory specially in AEPs. Can someone ...
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1answer
63 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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0answers
16 views

Confidence Interval of Information Entropy?

Information entropy, $IE$, is defined as: $$IE = \sum_{i} p_i log\frac{1}{p_i}$$ Where $p_i$ is the probability of event $i$ (and we are summing over all possible events). Let's say I have data ...
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Sequence of finite, positive and i.i.d random variables and limit of $\dfrac{S_{n+1}}{S_{n}}$

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of finite, positive and i.i.d random variables and let's call $\mu:=E(X_{1})>0$ and $S_{n}:=\sum_{i=1}^{n}X_{i}$. We know that ...
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2answers
25 views

Can someone help with this elementary example? [Absolutely Continuous Random Variables]

I'm having difficulty studying this part of the subject, because i can't get through this first example, can anyone help? Let $$X: U(0,1)$$ Find the distribution function of the following random ...
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34 views

Sum of not necessarily independent discrete random variables.

Let ${X_k}$ be a sequence of discrete random variables, where $P(X_k=k)=\frac{1}{k^2}$ and $P(X_k=0)=1-\frac{1}{k^2}$. Let $S_n=\sum_{i=1}^n X_i$. Does $\frac{S_n}{\sqrt{\log n}}\rightarrow 0$ (in any ...
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25 views

Sequence of independent events in a discrete probability space

Let $(\Omega, \mathcal{A}, \Bbb{P})$ be a discrete probability space. Let $A_1, A_2,...\in \mathcal{A}$ be a sequence of independent events with $p_n = \Bbb{P}(A_n)$. Then $$\sum_{n\in \Bbb{N}} ...
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36 views

Probability Theory Urn Question

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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18 views

Geometric elementary probability

I found the following question in a probability book I was solving:- $Q.$ A plane is ruled with parallel straight lines at equal distances of $2a$. A needle, $2l$ long$(l<a)$, is thrown on the ...
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Intuitive probability theory!?

Recently I saw something where someone had a paper with several lines on it and a needle. The length of the needle was the same as the distance between the lines. They then proceeded to say that when ...
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1answer
32 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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21 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
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20 views

What does it mean “rotationally invariant density”?

In the great answer given by the math.SE user @Tim, he does 2 hypothesis, on of the which ones is about the rotationally invariance of the density. Can you explain formally what does it mean? I do ...
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14 views

$A_n\sim XB_n$ and $A_n\leadsto N(0,V)$, can we say anything about $B_n\leadsto\ldots$?

Let $\sim$ denote an equality that holds up to a term that converges to $0$ in probability and suppose we have $$ A_n\sim XB_n $$ where $A_n$ is stochastic with dimension $r\times 1$, $B_n$ is ...
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1answer
23 views

PMF for K, the number of trails up to, but not including, the second success

I'm taking an MIT OCW course on Probability. Question: Al performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of three fair coins. ...
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1answer
26 views

Does $P(X>0)\geq\mathbb{E}[X]^2/\mathbb{E}[X^2]$ hold for a non-negative random variable $X$?

Is there a way to see if the inequality $$P(X>0)\geq\mathbb{E}[X]^2/\mathbb{E}[X^2]$$ is generally true for a random variable $X\geq0$ in $\mathbb{R}$?
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25 views

Finding an Expected Value

I encountered a simple calculation about finding an expected value on a composite function with Wiener process and Poisson process. For given $t>0$, assume $\alpha$ is arbitrary constant and ...
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1answer
48 views

Doob-style second moment martingale inequality

Let $\{X_k\}_{k=0}^{\infty}$ be a martingale, supposing $X_0 = 0$ and $E[{X_n}^2] <\infty$. Prove that $$P\left(\max_{1\le k \le n} X_k \ge r \right) \le \frac{E[{X_n}^2]}{E[{X_n}^2] + r^2}$$ ...