Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Prove X is a martingale

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by: $X_0 = 1$ and for $n \geq 1$ $X_{n+1} = 2X_n$ w/ prob 1/2 $X_{n+1} = 0$ w/ prob 1/2 and $\mathscr{F} = ...
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1answer
36 views

Coin-flipping experiment: the expected number of flips that land on heads

This question is from Sheldon M. Ross: Introduction to Probability Models which is about finding the expectation by conditioning. Question: A coin, having probability $p$ of landing heads, is ...
0
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1answer
29 views

Brownian Bridge equivalence of definitions

How can I verify the equality of the distributions arising from the two definitions of Brownian Bridge here: http://en.wikipedia.org/wiki/Brownian_bridge The two definitions are $W_t-tW_1$ and ...
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2answers
29 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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1answer
19 views

A question on semi-martingale and its variations

With probability one, paths of semimartingales have unbounded variation. What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of ...
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2answers
48 views

What is wrong with my method of finding the probability

One way to solve this and my book has done it is by : This is a well known way, but I have a different method, and it seems logical to me (but I don't know what the mistake is). And yes it's ...
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0answers
26 views

How can we prove a relation holds almost surely?

Let's we have a convex function $f:R \rightarrow R$, and g and x are random processes How can we prove the following relation holds almost surely? $\forall z\: \: f(z)\geq ...
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1answer
53 views

Does $x \perp (y,z)$ imply $x \perp y \mid z$?

Does $x \perp (y,z)$ imply $x \perp y \mid z$, where $\perp$ denotes stochastic independence? I was told it is true and the following is the proof (which I believe is wrong): We want to show that ...
3
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1answer
29 views

General Weak Law of Large numbers

I came across a question regarding the WLLN. Suppose for $X \geq 0$ , $\mathbb{E}[X] = \infty $ , $S_n = \sum_{i \leq n} X_i$, $X_i$ are iid copies of $X$ , and $\frac{\mathbb{E}[X \mathbf{1} _{X ...
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0answers
8 views

What does the independent statement $(x,y) \perp w \mid y$ mean and what does the “de-generate property” of probability mean?

What does the independent statement $(x,y) \perp w \mid y$ mean? I was guessing that it mean: $$p_{x,y,w|y}(x,y,w \mid \tilde{y})$$ where $y$ and $\tilde{y}$ are not necessarily equal. I interpret ...
2
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0answers
24 views

Does $(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp (Y,W) \mid Z$ hold?

I know that $ X \perp (Y,W) \mid Z \implies (X \perp Y\mid Z) \ \& \ (X \perp W \mid Z)$ but does the converse hold? i.e. does: $$(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp ...
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1answer
35 views

Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
0
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1answer
12 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
0
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2answers
36 views

Is this true: $\lim_{\lambda \rightarrow 0}E\left[ e^{-\lambda X} \right] = P\left(X < \infty \right)$?

Is it true that $\lim_{\lambda \rightarrow 0} E\left[ e^{-\lambda X} \right] = P\left(X < \infty \right)$ ? I've seen this in a few places online but I can't seem to be able to find a proof online ...
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0answers
25 views

Subgaussian bounds for $X$ imply subgaussian bounds for $X-E(X)$

A random variable $X$ is called sub-gaussian, if there exist positive constants $C,c$ such that for all positive $\lambda$ we have $$P(|X|\geq \lambda)\leq Ce^{-c\lambda^2}.$$Now I'm reading a text, ...
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1answer
34 views

A Question about an example in Durrett's Probability textbook

I was reading an example in Durrett's book: Probability : Theory and Example, 4th edition (pdf verison) (Example 3.4.7, p.112) The scenario is as follows: Define $Y_1,Y_2...$ be independent ...
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1answer
17 views

Jump Set v. Range of Randome Variable

What is the difference between the range of a random variable X, and its jump set? I know that they are not equivalent sets, e.g. for a continuous RV, the range is $(- \infty , \infty)$, but the jump ...
4
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1answer
31 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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2answers
26 views

Distribution of a distance between random numbers

I'm working on a problem in which I came to a question concerning distribution law of a result of operations on random variables. I will ask a simple question and hope to understand the concept from ...
0
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1answer
22 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...
2
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1answer
18 views

Finding expectation and variance for multiple items when given for 1 item

The problem I have here is that I know this is a normal distribution question, but I don't know how to find the Variance And Mean for 50 items, they have given for 1 item My book simply says : ...
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1answer
21 views

A problem on almost sure convergence of an average

I have the following exercise: Let $X_1, X_2 \ldots$ be such that $$ X_n = \left\{ \begin{array}{ll} n^2-1 & \mbox{with probability } n^{-2} \\ -1 & \mbox{with probability } ...
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0answers
16 views

Definition of n independent event and example

Given a finite set of events $A_1,\dots,A_n$, the events are said to be independent if and only if for any subset of indices $I$ we have: $$\mathrm{P}\left(\bigcap_{i\in I} A_i\right)=\prod_{i\in I} ...
4
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0answers
73 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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0answers
24 views

Probability of an event happen using a uniform number?

I am starting to work in a mathematical modeling in agriculture field, and I am not a statistician or mathematician. Could you please help me to explain this? I have a curve of "probability of ...
0
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0answers
41 views

Conditional mean: E(Y|x)

Please help.I am not sure with my answer.Anyways, the problem goes this way: Find the conditional mean of $Y$ given $X=x$ ,$E(Y|x)$, if X and Y have the joint pdf of $f(x,y)=21x^2y^3, ...
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1answer
34 views

Probability of $P(X_1<X_2|X_1<2X_2)$

Find the $P(X_1<X_2|X_1<2X_2)$ Given: $$f(x) = e^{-x}, \qquad 0<x<\infty$$ zero elsewhere. The rvs have same pdf and they are independent variables. Here is my attempt: ...
0
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1answer
62 views

On a decomposition of a conditional distribution

I am trying to make some sense out of equation (7) in the recent paper of Peter van Leeuwen: "Representation errors and retrievals in linear and nonlinear data assimilation" ...
0
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1answer
16 views

Distribution function and probability of random variable R with density function $f(x) = 1/2e^{-|x|}$

So we have a random continuously variable $R$ with density function $f_R(x) = 1/2e^{-|x|}$. First I need to sketch the distribution function of $R$. So $$F_R(x) = \int_{-\infty}^x f_R(x) dx$$, but ...
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3answers
56 views

question at Probabilities [closed]

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the ...
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2answers
31 views

How is the binomial distribution connected with the theoretical approach to probability?

I've been told the theoretical approach to probability is defined as follows $$\operatorname{Pr}(\textsf{something})=\frac{\textsf{Favorable events}}{\textsf{possible events}}$$ This has to be ...
0
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1answer
21 views

Are 2 dependent probabilities always disjunct?

If 2 probabilities are disjunct then they are not independent. Are any 2 dependent probabilities also disjunct? Modus ponens for instance are 2 dependent and mutually exclusive events (A and B) where ...
2
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0answers
25 views

Is there a continuous version of the Borel-Cantelli lemma?

Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limes ...
2
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1answer
21 views

About modifications of right-continuous stochastic processes

Lemma : Let $X$ and $X'$ be two right continuous(or left continuous) processes defined on the same probability space $(\Omega,F,P)$ be a modification then the two processes are indistinguishable. ...
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0answers
24 views

Convolution of two random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
0
votes
1answer
33 views

conditional distribution of X, given Y=y and Z=z, and compute E(X|y,z)

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere. Find the conditional distribution of $X$, given $Y=y$ and $Z=z$, and compute $E(X|y,z)= ...
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1answer
24 views

Portmanteaus Theorem and Lipschitz functions

I was looking at some basics in the theory of convergence of distributions, e.g. Portmanteaus Thm. One part of it asserts that it suffices to check $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for bounded ...
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1answer
37 views

Probability of $P(X_1 X_2\le 2)$

What is the probability of $P(X_1 X_2\le 2)$. Both variables are independent and each has the probability density function $f(x)=1, 1<x<2$, zero elsewhere. First I would like to assume that the ...
2
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1answer
28 views

Expectation of proportion random variable

Suppose that $X$ and $Y$ are two random variable. $X$ and $Y$ are independent. We also have $f$ and $g$ are two continuous functions on $\mathbb{R}$. Is the following equation true? $$ ...
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0answers
10 views

Calculate Expectation of points in a homogenous poission process with parameter $\alpha $ as a renewal process?

If a poisson process $N $ on $[0, \infty ) $ has rate $\alpha $ (ie $E N(A)=\alpha m(A) $, $m $ lebesgue measure ) can its points be represented as occurences in a renewal process with interarrival ...
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46 views

How to prove that $X+Y \mod p$ is indpendent from $X$ if $X$ and $Y$ are independent?

We have a group $\mathbb{Z}_p$ and some random variable $X$ and $Y$ with this domain. We have that $Y$ is chosen uniformly at random, thus each element from $\mathbb{Z}_p$ with probability ...
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1answer
26 views

Finding a CDF from the PDF of another Random Variable

Given a function $$ Y=ae^x $$ With the distribution: $$ f_X(x)=be^{-bx} \,\,\,\, x\geq0 $$ Show that the cumulative distribution function is: $$ F_Y(y)=1-(y/a)^{-b} \,\,\, y\geq c $$ My approach ...
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1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
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1answer
16 views

Using Monotone Convergence Theorem to extend a result involving random variable

We assume that for a non-negative, bounded, continuous random variable we have $$ E[X]=\int_0^\infty P(X>x) dx $$ Now the task is to extend this result to non-negative, continuous random variables ...
2
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1answer
40 views

Alpha mixing property of a $\mathbb{R}^d$ valued Stochastic Process

In statistics and probability literature, a strictly stationary stochastic process $\{X_t\}\in\mathbb{R}$ is called $\alpha$-mixing if $\alpha(n)=\sup_{A\in\mathcal{F}_{-\infty}^{0}, ...
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2answers
76 views

Best advanced book in probability theory?

I have read real and complex analysis, and Probability theory by Feller(Vol1 & 2). What is the next book I should read to go deeper into subtopics of probability theory? Thank you!
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1answer
10 views

Additivity of the Chi-squared distribution

Can additivity of the $\chi^2$-distribution be used when substracting variables? For example: Does X-Y=Z where X~$\chi^2$(3), Y~$\chi^2$(1), Z~$\chi^2$(2)?
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30 views

Creating a random network (graph) with a $\textbf{random}$ number of vertices and given degree distribution

I was trying to find an answer to my question on google scholar, however I didn't find anything that is close to what I am looking for. I would be very grateful for your help. There is a theory of ...
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0answers
41 views

Lindeberg condition of order $2k$

Prove that if $ (X_n),\: n \ge 1$ are independent r.v.s with $ E X_n = 0, E X_n^2 = \sigma_n^2$ which obey central limit theorem and $$\lim_{n\rightarrow\infty}E\bigg(\frac{S_n ...
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0answers
31 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...