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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
7 views

Expectation with respect to empirical distribution

Let $(\Omega,\mathcal{A})$ be a measure space and $X$ a random variable with distribution $P$. The expectation of some measurable function $g$ with respect to $P$ is $$ \mathbb{E}_P[g(X)] = ...
1
vote
3answers
17 views

Family of functions that are bounded in $L^1$ but *NOT* Uniformly Integrable

I'm having a difficult time constructing a counter example to this. My intuition (sloppily) is to construct a family of functions {$X_n$} that have Dirac pulses at $n$ and $-n$. Such that $\sup_n \Bbb ...
1
vote
0answers
7 views

Conditioning a Brownian Bridge

Brownian Bridge: Consider the standard Brownian $X(t)$ motion conditioned to land at $b$ at time $1$. This means that for every trajectory {$(t,X(t)), 0 \leq t \leq 1$} of this process, its initial ...
0
votes
0answers
5 views

Weak Law of Large Numbers for uniformly integrable, independent random variables

On page 58-59 of the notes by Knill (found here :http://www.math.harvard.edu/~knill/teaching/math144_1994/probability.pdf ) there is a version of the WLLN whose proof I have trouble understand. On ...
1
vote
1answer
17 views

Evaluating probabilities with joint density.

I have a density function $f(x,y)=\alpha x^2y^2$ for $y\in (0,1)$ and $x\in (-y,y)$. To find $\alpha$ I evaluated $$\alpha \int_{-y}^y\int_0^1 x^2y^2 \ dydx=1$$ and ascertained $\alpha=9/2y^3$. How ...
1
vote
0answers
20 views

PDF of $|X(t)| =| e^{j\omega_c t}+W(t)|$

let $X(t) = Ae^{j\omega_c t}+W(t)$, where $W(t)$ is a gaussian process that follows the statistics $W \sim \mathcal{CN}(0,\sigma^2)$ and $\omega_c$ denotes the carrier pulse frequency and $A$ is a ...
0
votes
1answer
21 views

A woman has n keys

A woman has n keys, of which one will open her door. After trying one she discards it and tries again if it does not work. What is the expected number of attempts needed? Its straight forward to see ...
1
vote
0answers
24 views

Independent random variables and integrability

This is a problem that I am stuck at. I think I have to prove the hint first. But I can't find a way to prove the 'only if' part of the hint. (the 'if' part is just manifest). Could anyone help me ...
0
votes
2answers
39 views

Given the probability distribution of X, whats the PDF of X²?

Let's say we have a random variable $X$ with a certain probability density function $f_x(x)$. 1) How should I find out the PDF of the random variable $X^2$? Problem background: $X_1 = s_1 + W$, ...
1
vote
2answers
172 views

Convergence in Probability

Consider a sequence of $N$ Bernoulli trials with, with probability of success denoted by $p$, and let $X$ be the number of successes. Show that as $N\rightarrow\infty$, $\frac{X}{N}$ converges in ...
2
votes
1answer
31 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 ...
0
votes
1answer
24 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 $Ef(Y_n) \rightarrow Ef(Y)$ ...
1
vote
2answers
19 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 ...
0
votes
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 ...
1
vote
1answer
35 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 ...
2
votes
1answer
16 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 ...
1
vote
0answers
30 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
votes
1answer
27 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
votes
1answer
76 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?
1
vote
0answers
13 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 ...
0
votes
1answer
23 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
votes
1answer
34 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
votes
0answers
16 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 ...
-1
votes
0answers
17 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 ...
1
vote
2answers
36 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} ...
1
vote
1answer
16 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, $\Delta (X)$ ...
0
votes
1answer
47 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. ...
1
vote
2answers
30 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 ...
0
votes
1answer
23 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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
16 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 ...
5
votes
2answers
70 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 ...
0
votes
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 ...
0
votes
1answer
13 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 $J$ ...
0
votes
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 $Y \in \{0,1\}$ ...
1
vote
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 ...
4
votes
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 ...
0
votes
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 ...
0
votes
1answer
36 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
0answers
10 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)| ...
1
vote
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
votes
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:- ...
0
votes
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. ...
1
vote
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 | ...