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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17 views

Martingale Convergence Theorem for non-negative supermartingales: Why is limit non-negative?

Consider a supermartingale $(X_n)$ that is non-negative. The martingale convergence theorem states that $X_n \rightarrow X$ P-a.s. with $X \geq 0$ and $E[X] \leq E[X_0]$. Why can we conclude that $X ...
1
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1answer
65 views

Simple CDF Computation for Products of Random Variable

Let $X(k)$ be i.i.d random variable governed by uniform distribution $[-1,1]$ for $k=0,1,2,...N$. I would like to compute the following CDF $$ P\left( {\prod\limits_{k = 0}^{N - 1} {(1 + X(} k)) ...
2
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1answer
23 views

What is the probability that there are $k$ people between $A$ and $B$ sitting around a circle?

I have $n$ people seated around a circular table and $2$ of the people are $A$ and $B$. What is the probability that there are $k$ people between $A$ and $B$? I have tried noting that the total ...
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0answers
33 views

3d symmetric random walk passes infinitely through any particular line

I'm trying to solve problem 27 from Chapter XIV An Introduction to Probability Theory Volume I by William Feller, ...
2
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1answer
37 views

Calculate the probability, that a man repair 20 machines in 8 hours. It is correct my work?

The problem statement said: The servicing of a machine requires two separate steps, with the time needed for the 1st step being an exponential random variable with mean 10 minutes and the ...
3
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3answers
83 views

Conceptual Statistics. Define for this problem, population, Samples and Estimators, and when is Normal Dist?

Students in Stanford are supposed to spend on average 3 hours of time per week for every credit hour they take. Last year, 263 randomly selected seniors were contacted and asked how much total time ...
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0answers
39 views

Distance between Brownian Motion and scaled Gaussian random walk

I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact: If $Z(t)$ is a standard Brownian Motion and ...
2
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0answers
62 views

Will the generated sigma algebra have this property?

Lets say you have a measurable space $(\Omega, \mathcal{A})$. And a measurable function $X: (\Omega, \mathcal{A})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. We then know that for the sigma ...
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0answers
52 views

Infinite sum of indicators almost sure convergence

Let $S_{n}:= \sum_{i=1}^{n} X_{i,n}$ where for each $n, X_{1,n}, X_{2,n},..., X_{n,n}$ are sequences of independents r.v.'s. $$X_{i,n}=\begin{cases}1, & \text{with probability }p_n\\0,& ...
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1answer
45 views

What is the probability that the upturned faces of three fair dice are all of different numbers?

Three fair dice are rolled ($6$ sides). What is the probability that the upturned faces of the three dice are all of different numbers? I got that the number of possible outcomes total is $6^3$ ...
1
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0answers
21 views

part of proof about stochastic processess

I need help proving a part of a proof. The exercise in the book(exercise 3.14 in Bernt Øksendals: Stochastic Differential equations) is given as to prove something, but I will only talk about the ...
3
votes
1answer
40 views

Question about weak convergence of random variables

When you start to learn probability theory, for instance the central limit theorem, you learn about convergence in distribution $X_n\to X$ (where, say, both $X_n$ and $X$ are $\mathbb R$-valued random ...
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0answers
18 views

Probabilities of events involving random variables with random indices

Let, on some probability space $(\Omega,\mathscr F,\mathbb P)$, $(Z_t)_{t\geq0}$ be a real-valued (measurable) stochastic process. Moreover, let $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$ be ...
1
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1answer
29 views

Integral estimate for positive part

I'm somewhat stuck on understanding, what seems to be a kind of elementary estimate. Let $X\geq 0$ with $\mathbb{E}(X)=1$. $f:\mathbb{R}^+ \to \mathbb{R} $ and $f(x)\geq-c>-1$. Let $A$ be some ...
0
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0answers
21 views

When $X_{n\wedge N}$ converges to $X_N$ in probability for martingale $X_n$ and stopping time $N$?

Suppose $\sigma$-algebras $\{\mathcal{F}_n\}$ is a filtration and random variables $\{X_n\}$ are adapted to $\{\mathcal{F}_n\}$. $N$ is a stopping time w.r.t $\{\mathcal{F}_n\}$. If $(X_n, ...
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1answer
19 views

Probability with Indicators Textbooks

I am new to using indicator functions (although I am quite familiar with undergrad-level probability and what an indicator function is). I am trying to relearn probability using indicator functions ...
3
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2answers
53 views

Conditional expectation of Poisson r.v. $X$ given $X$ is even?

We have a random variable $X$ that is poisson distributed with $\lambda$. We wish to show: $$E[X\mid X \text{ is even}]=\lambda \frac{1-e^{-2\lambda}}{1+e^{-2\lambda}}$$ So far, I have that $P(X ...
1
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1answer
19 views

Conditioning Expectated Value on Independent Random Variables

Let $X_1$ and $X_2$ be independent random variables. Let $Y$ be a random variable such that $E(Y\mid X_1)=0$ and $E(Y\mid X_2)=0$. Under what conditions is it true that $E(Y\mid X_1,X_2)=0$?
1
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1answer
26 views

CDF for Laplace distribution

According to the Wiki article on the Laplace distribution, $$F(x)=\int\limits_{-\infty}^x f(u)du=\begin{cases} \frac{1}{2}\exp(\frac{x-\mu}{b}) && \text{if }x< \mu \\ ...
2
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1answer
26 views

Is difference of two independent Gaussian R.Vs and their sum independent?

I have been trying to answer a question I have been carry on from probability course. I'd appreciate if anyone can help me. Suppose we have two independent Gaussian distributions, both zero-mean with ...
3
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1answer
22 views

How to show that Cantelli's inequality has no better result

Cantelli's Inequality states that for a random variable $X$ with mean $μ$ and variance $\sigma^2$: $$ P(X-μ\geq \alpha)\leq\frac{\sigma^2}{\sigma^2 + \alpha^2} $$ Now, I read that if I consider a ...
1
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1answer
24 views

Is the variance of a random variable finite if and only if the first two moments of the random variable are finite?

I don't know measure theory very well, and I'm not looking for a very detailed proof. Suppose $X \in \mathcal{L}^2$ (I believe this means that $\mathbb{E}[X^2]$ exists, which implies that ...
1
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1answer
25 views

Is it possible to show $P(|X-μ| \geq \alpha)\leq\frac{2\sigma^2}{\sigma^2 + \alpha^2}$ from Cantelli's inequality?

I know that Cantelli's Inequality states that for a random variable $X$ with mean $μ$ and variance $\sigma^2$: $$ P(X-μ\geq \alpha)\leq\frac{\sigma^2}{\sigma^2 + \alpha^2} $$ Here, I am trying to ...
0
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1answer
50 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
4
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0answers
44 views

An inequality regarding the expectation of a transformed random variable

For any given nonnegative random variable $X$ and $0 < \rho \leq 1$, I define the following object. $$\pi_{\rho}(X) = \int_0^{\infty}(P\{X > t\})^{\rho}\, dt$$ The inequality that I want to ...
3
votes
1answer
35 views

Application of Doob's inequality

Suppose that $X_n$ is a martingale with $X_0 = 0$ and $EX^2_n < \infty$. Show that $$P\left(\max_{1\leq m \leq n} {X_m} \geq \lambda\right) \leq \frac{EX^2_n}{EX^2_n+\lambda^2}$$ by using ...
2
votes
1answer
28 views

Convergence in probability implies Fatou's lemma?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable ...
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1answer
25 views

Almost everywhere equality of r.v.'s , based on information on mean values.

Let $X, Y$ be two random variables on a probability space $(\Omega, \mathcal{F},P)$, where $\mathcal{F}=\sigma(\mathcal{E})$. We assume that: $\mathcal{E}$ is closed on intersections, i.e. $A\cap ...
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0answers
24 views

Prove the probability that X takes on an even value for a Poisson R.V.? [duplicate]

We have a random variable $X$ that is distributed according to the Poisson distribution with $ \lambda$, such that $P[X=k]=e^{-\lambda}\frac{ \lambda^k}{k!}$ for $k=0,1,\dots$. We want to show that ...
1
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1answer
31 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
2
votes
2answers
57 views

If $X_n = n \ \text{w.p.} \ \frac{1}{n}$ and $X_n = 0 \ \text{w.p.} \ 1-\frac{1}{n}$, what is $E\left[X_n \mathbf{1}_{(|X_n|>t)}\right]$?

If we have a random variable $X_n$ such that $$X_n=\begin{cases}0, &\text{with probability }1-\frac1n\\n, &\text{with probability }\frac1n\end{cases}$$ How can I find $E\left[X_n ...
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0answers
47 views

Understanding the concept of measurability of random variables

If a random variable $X$ is $\mathcal{F}_{t_0}$-measurable, where $\{ \mathcal{F} \} _{ t \geq 0}$ is an underlying filtration, does that mean that from the time $t_0$ onwards, the random variable $X$ ...
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0answers
59 views

Markov Chain Probability Limit [on hold]

Show that the Markov chain on the state space $S=\{0,1,...,n \} $ with transition matrix: $$ P(k,l) = \binom{n}{l} \left(\frac{k}{n}\right)^{l} \left(\frac{n-k}{n}\right)^{n-l} $$ is such that, ...
1
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2answers
32 views

Q: Proof of the linearity of expectation of random variable on page 38 of Shiryaev's Probability

This is about the proof of the property of expectation of random variable: $\mathbb{E}(a\xi+b\eta)=a\mathbb{E}\xi + b\mathbb{E}\eta.$ (finite probability space) The proof on the book is as ...
2
votes
1answer
60 views

Prove that the stochastic process can not have continuous paths.

This problem is about stochastic processes, but what I really need help with is using the dominated convergence theorem in the end: What I need to prove is that a stochastic process having these ...
0
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0answers
28 views

How to find $\limsup_{a \to 0} E[V | aV+W]$

How to find \begin{align} \limsup_{a \to 0} E[V | aV+W] \end{align} where $V$ and $W$ are independent. We can assume that $E[V^2], E[W^2] <\infty$. I think the following must \begin{align} ...
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1answer
29 views

If we have events $E_k, k \in \mathbb{R}$ such that $P(E_k) = 0$ for all $k \in \mathbb{R}$, what is $P(\cup_{k \in \mathbb{R}}E_k)$ equal to?

Here, we have an uncountably many number of sets. In this case, we cannot use Boole's inequality to show it is equal to 0 as in the countable case, but I am not sure whether and what the value is. Can ...
2
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1answer
20 views

A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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1answer
15 views

Spatial Arrangement of Permutational Probability

I have 4 black balls and 4 red balls, I need to fill 8 baskets with one ball each. How many different arrangements can I have such that each arrangement is unique. So if black balls are represented as ...
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0answers
34 views

If $X$ is a discrete random variable, then $\mathcal{X}$, the range of $X$, is countable.

$F_{X}$ denotes the CDF of a random variable $X: S \to \mathbb{R}$ ($S$ the sample space). I'm not caring about the $\sigma$-algebra technicalities for now. My definition of a discrete random ...
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0answers
16 views

Is $\arg \max_{(t_1,t_2), t_1\geq t_2} (\max_i (E(x_i\mid x_1 \geq t_1, x_2 \geq t_2) +\cdots)$ unique for given distr $(x_1,x_2) \simeq F([0,1]^2)$?

Ingredients and notation: Given is a joint distribution $F$ of two variables $(x_1,x_2)$ where $x_i \in [0,1]$, with a strictly positive joint density $f(x_1,x_2) \in C^1$, that is continuously ...
1
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1answer
25 views

Sum of probabilities versus probability of sum

Consider two real-valued random variables $X,Y$ defined on the probability space $(\Omega, \mathcal{F}, P)$. let $\epsilon>0$. Could you help me to show that $$ P(|X|+|Y|\geq 2\epsilon^2)\leq ...
2
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2answers
46 views

Rigorous Probability/Statistics Book reference?

Im wondering if anyone could recommend a book (or a few books) about statistics/probability for someone at the advanced undergraduate level who has taken some real analysis (at the level of baby ...
2
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0answers
56 views

How to understand $E(X\mid B)$ in the measure theory way

From undergraduate probability course, we learn $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ given $P(B)>0$. And we learn that if $(X,Y)$ has a joint density $f(x,y)$, we can calculate marginal density ...
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0answers
19 views

Symmetric random walk ergodic [closed]

Consider a symmetric random walk on $\mathbb{Z}/m \mathbb{Z},$ i.e. we start in some state $[k]$ and then propagate with equal rates either to $[k+1]$ or $[k-1]$ and so on. How do I show that this ...
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0answers
21 views

Weak convergence time-continuous random walk

I was wondering whether the time-continuous random walk on $\mathbb{Z}$ that I want to denote by $X:[0,\infty) \rightarrow \mathbb{Z}$ with $X(0)=0$ a.s. and transition rates(NOT probabilities) ...
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1answer
25 views

Asymptotic distribution of a measure of homogeneity

For an exam preparation I'm trying to solve the following question, but I get stuck. The question is One measure of the homogeneity of a multinomial population with $k$ cells and probabilities ...
2
votes
1answer
38 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that ...
2
votes
1answer
31 views

linear combination of infinitely divisible random variables

If $X$ and $Y$ are real valued random variables with infinitely divisible distributions, does $aX + bY$ also have an infinitely distribution ($a, b \in \mathbb{R}$). I've seen this stated in several ...
0
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0answers
29 views

Getting the independent variables from dependent variables. [duplicate]

This question is related to the solution in the answer here: Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. Quick description of my problem: Let ...