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 [tag:probability-...

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13 views

For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely

Question in the title: For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means.....
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15 views

A result about weighted-sum of uniform random variables

Let $a_1,\ldots,a_m \in \mathbb{Z}$ and $U_1,\ldots,U_m$ be independent uniform random variables taking valules in $[0,1]^d$. Let $\mathcal{Z}$ be the support of the random variable $\sum_{i=1}^m a_i ...
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0answers
15 views

Predict the daily usage of Bandwidth of a Network

Context: I want to predict the daily usage of bandwidth of a network (consists a number of users) based on previous use . For example, I want to predict the amount of bandwidth during 8 pm to 9pm ...
14
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1answer
884 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
3
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2answers
42 views

How do I prove that for a random variable $X$, we have $P(X \le a) \le p$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$, $\mu = \mathbb{E}(X) = 2$, and $\max(X) \le 10$, (or $P(X \ge 10) = 0$). How can I prove the following? $$P(...
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0answers
19 views

Example of Markov property

I am reading Durrett's Probability: Theory and Examples, and trying to understand its context about Markov Chains. It is not hard to understand the theorem and proofs but when it comes to concrete ...
0
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0answers
38 views

How can I prove that for a random variable $X$, we have $P(X \le \mu) = P(X \ge \mu)$?

Specifically, suppose that $X$ is a random variable with properties $\mathrm{Var}(X) = 9$ and $\mu = \mathbb{E}(X) = 2$. How can I prove the following? $$P(X \ge \mu) = P(X \le \mu)$$ It is also ...
2
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1answer
16 views

When is the sum of uncorrelated (not necessarily with the same distribution) r.v.'s bounded in Probabilty?

Let $v_{i},\;i=1,\cdots,N$ be such that $E\left(v_{i}\right)=0$, $E(v_{i}^{2})=1$ and $E\left(v_{i}v_{j}\right)=0\;for\;i\neq j$. So $v_{i}$'s are mean zero with unitary variance, uncorrelated and ...
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1answer
15 views

Chi-Squared Distribution

Let $Z_1, Z_2, Z_3$ be independent standard Normal R.V.'s. Which of the following has a Chi-Square distribution with 1 degree of freedom. $$ \begin{align} A) & & & \frac{Z_1^2, Z_2^2}{2} ...
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1answer
67 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all $n\...
2
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1answer
31 views

Is it true that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$?

I was wondering if we can show that $E(X_1\mid X_1+X_2=k+1)−E(X_1\mid X_1+X_2=k)≤1$ in general? Here $X_1$ and $X_2$ are independent but may not follow the same distribution. Any hint is much ...
2
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0answers
33 views

What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
4
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1answer
26 views

$E(X_1|X_1+X_2=k)$ increases with $k$?

$X_1$ and $X_2$ are independent, but they may not follow the same distribution. I want to know whether $E(X_1|X_1+X_2=k)$ increases with $k$. I guess this is correct, but is there a proof or counter ...
3
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1answer
125 views

Are there any modern mathematicians whose research interest is in “Probability Theory”? [closed]

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
2
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1answer
472 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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0answers
23 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
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0answers
9 views

Gram matrix for a random variable vector space with inner product?

I am wondering if it is possible to construct a list of binary valued random variables, $\{\bf{X}_1,\bf{X}_2,\bf{X}_3\}$ and define a Gram-like matrix like \begin{bmatrix} \langle\bf{X}_1,\bf{X}_1\...
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0answers
14 views

Prove $\frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\rho^2}}$ [on hold]

Let $X_1,X_2$ have a bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Show that $$ \frac{\partial}{\partial\rho}P(X_1 > 0, X_2 > 0) =\frac{1}{2\pi\sqrt{1-\...
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0answers
35 views

Discrete random variable whose cdf is not a step function

Let, $(\Omega,\mathcal{F},P)$ be a probability space and $X:\Omega \rightarrow \mathbb{R}$ be a random variable. Let $F_{X} (x)$ be the cumulative distribution function of $X$. Show that if $F_{X} (x)$...
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0answers
16 views

Probabilistic Method/Model for Traffic Flow

Context: Given a network system or a traffic system with some value related to the system. Question: Which probabilistic methods, model, distributions are used frequently to predict a event (for ...
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1answer
387 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
2
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1answer
26 views

Two notions of square-integrability

It seems to me there are two notions for random variables / processes which get labeled square-integrable: $EX^2_t<\infty \; \forall t$ $E \int^t_0 X_s^2 \; ds < \infty \; \forall t$ I ...
0
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1answer
38 views

What does a random variable 1 with subscript [0,1/2] mean?

I came across the following notation that I cannot follow: $1_{[0,1/2]}$ It is supposed to be some kind of random variable (or just an event? not sure) It is hard to google this, too. What does such ...
2
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1answer
37 views

determine the distribution of the random variable $Y=\Sigma_{k=1}^{\infty}kX_k$

Fix $p \in (0,1)$ and consider independent Poisson random variables $X_k$, $k \geq 1$ with $\mathbb E[X_k]=\frac{p^k}{k}$. Verify that the sum $\Sigma_{k=1}^{\infty}kX_k$ converges with probability ...
0
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0answers
21 views

Variance: logical/mathematical meaning [duplicate]

$$\operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right]$$ Is the formula of variance. But if you think of it, you can assume that square was introduced just that something other than ...
43
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8answers
10k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
2
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1answer
29 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
4
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1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
4
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1answer
51 views

conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
0
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1answer
36 views

Convolution of a function with itself n times convergence to bell curve [duplicate]

If we have a piecewise function defined as $f(x) = \begin{cases} 1, & \text{0 $\le$ $x$ $\le$ 1} \\ 0, & \text{otherwise} \end{cases}$ Explain how the convolution of $f$ with itself for $n$ ...
1
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1answer
32 views

Moment generating function of $X+Y$ using convolution of $X$ and $Y$

Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused ...
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0answers
23 views

Convergence of expectations of a sequence of exponential random variables.

Suppose $\{X_n\}$ is a sequence of exponentially distributed random variables, where $X_n$ has mean $1/\lambda_n$. Suppose that $\lim_{n\to\infty}\lambda_n = \lambda>0$. Let $X$ be exponentially ...
3
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4answers
92 views

Find the value of $\mathbb{E}(X_1+X_2+\ldots+X_N)$ of i.i.d random variables $X_i$s.

Let $ X_1,X_2,X_3 ,…$ be a sequence of i.i.d. random variables with mean $1$. If $N$ is a geometric random variable with the probability mass function $\mathbb{P}(N=k)=\dfrac{1}{2^k}$; $k=1,2,3,\...
3
votes
1answer
87 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
1
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0answers
35 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
1
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1answer
51 views

Expectation of $|H - T|$

Using binomial approximation to normal distribution, find the expectation of $|H-T|$ where the $H,T$ are heads and tails of a fair coin and the number of tosses is large. Can anyone please tell me, ...
3
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1answer
29 views

Explanation of Aumann's “agreeing to disagree” in modern notation

I'm attempting to understand Aumann's classic 1976 paper Agreeing to Disagree, which claims, under certain assumptions, that if two Bayesian agents share knowledge of each others' posteriors then they ...
2
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1answer
36 views

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to?

Let $X$ be a standard normal random variable. Then, $ P(X<0\mid |[X]| = 1)$ is equal to- $\frac{\Phi(1)-\frac{1}{2}}{\Phi(2)-\frac{1}{2}}$ $\frac{\Phi(1)+\frac{1}{2}}{\Phi(2)+\frac{1}{2}}$...
0
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2answers
31 views

basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
0
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1answer
11 views

Notation clarification in Schilling's Brownian Motion

In Chapter 1's Problems, Problem 1(b), we are given that $X,Y\sim \beta_{1/2}:=\frac{1}{2}(\delta_0+\delta_1)$ are Bernoulli random variables. How am I to interpret this? That the probability of ...
4
votes
1answer
1k views

Inverse Mills ratio for non normal distributions.

We have the well known result of the inverse Mills ratio: $$ \mathbb{E}[\,X\,|_{\ X > k} \,] = \mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
0
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1answer
52 views

Show that $\frac{S_n}{n}\to 0$ in probability if $s<\frac{1}{2}$

Let $s\in\mathbb{R}$ and $X_1,X_2,\dots$ be independent random variables and with distributions: $$P(X_n=n^s)=P(X_n=-n^s)=\frac{1}{2}$$ Let $S_n=X_1+\dots+X_n$. Show that $$\frac{S_n}{n}\to 0 \text{ ...
1
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1answer
22 views

Find variance of happy people sitting at $n$ regular polygon table

Let table has shape of $n$ regular polygon and at each side is sitting one person. Each person is flipping a fair coin once (results of $n$ independent tosses are independent). Person is happy iff he ...
2
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1answer
50 views

Find the limit of $P(\bar{X_n}\leq 1.8)$ for i.i.d random variables $X_i$s of known distribution

Let $X_1,X_2,…$ be a sequence of independent and identically distributed random variables with $P(X_1=1)=\frac{1}{4}$ and $P(X_1=2)=\frac{3}{4}$. If $\bar{X_n}=\frac{1}{n}\sum_{i=1}^{n}X_i$, for $n=...
2
votes
1answer
67 views

Sequence of Erdos-Renyi random graphs convergent with probability 1

Definitions Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph ...
2
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0answers
29 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
0
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0answers
11 views

CGF determines the distribution

It is well known, that if the domain of the mgf of a random variable $X$ contains an interval around zero, then the distribution is completely determined by its moments. However consider a Levy-...
1
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0answers
35 views

Probability question, can I reset the window or not

There is a wall street banker. The banker invests in a kind of share called as options. The main features of this share is as follows: You make a bet with a specified amount of information as to ...
3
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0answers
16 views

Information matrix for a Student's T distribution

I'm reading a paper from Creal, Koopman, Lucas "Univariate Generalized Autoregressive Score Volatility Models" and I'm stuck with this computation. $$ -\operatorname{E}_{t-1} \left[ \frac{\partial^...
4
votes
1answer
46 views

What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...