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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Probability/Statistics

Let {Xr : r>=1} be independently and uniformly distributed on [0,1]. Let 0 N=min{n>= 1 : X1 + X2 +....+Xn> x} Show that P(N>n) = $ \frac{x^n}{n!}$ and hence find the mean and variance of N. I cant ...
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11 views

Equivalent condition for $ (X_n, \mathcal{F}_n) $ to be a martingale

I've encountered an interesting problem and am not quite able to solve it. It is to prove the following statement ($ X_n $ denotes a sequence adapted to a filtration $\mathcal{F}_n) $: $$ ...
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0answers
12 views

Why is “having countably many open rays” a measurable condition

In discussing Bernoulli($p$) percolation on a tree, sometimes one asks the question of what the probability is that there are countably infinitely many rays containing only open edges. I don't see ...
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10 views

a way to compute energy of a flow on transient trees

Let $T$ be a rooted tree that is also a transient electrical network (so effective resistance from root to infinity is finite) but with recurrent rays. (So effective resistance root to infinity along ...
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0answers
18 views

Repeated coin flips probability [on hold]

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated T times. Assume the $k$'th flip for all possible $k$ values ($1 \le k \le L$) among all experiements. If the head ...
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0answers
23 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
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1answer
36 views

What does it mean that an expected value does not exist?

$X$ is a random variable with pdf $f$ and $g: \mathbb R \to \mathbb R$ is a measurable function. Before I start operating with $E[g(X)]$ I need to show that it exists. What does it take to show it? ...
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0answers
5 views

M/M/1 Queue Under Heavy/Light Loads

Given, $\rho \approx 1$ (heavy-load), for the open M/M/1 queue, can we consider the sequence of response times an approximate renewal process? That is, the sequence of measured response times from a ...
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2answers
69 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
3
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1answer
38 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
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0answers
26 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
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33 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
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1answer
37 views

Give simpler examples of $L^p$ unbounded yet converging martingales

I read the following example in the book Counterexamples in Probability and Real Analysis by Gary L. Wise and Eric B. Hall: Does anyone know simpler examples? I do have one! I would be glad to ...
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1answer
13 views

Question about “integrable” random variable

I was reading the definition of Markov's Inequality on Wikipedia and it says If $X$ is any nonnegative integrable random variable and $a > 0$, then $\mathbb{P}(X \geq a) \leq ...
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2answers
26 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
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1answer
25 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
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0answers
31 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
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0answers
27 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
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20 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[lim_{n \to ...
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1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
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21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
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1answer
22 views

ways to choose 4 people including two or more male from N people with 1/3 male 2/3 female

A sample of four people is randomly drawn from a population of N > 4 people.Assume that 1/3 of the total population is male, and 2/3 is female. (To simplify things, let’s assume that N is always ...
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25 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
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1answer
24 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
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1answer
15 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
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1answer
13 views

Showing that the union of two algebras will give us a disjoint finite union of sets.

Suppose we have a space $X$ that could be anything and $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ are algebras. How would I show that that algebra generated by $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ ...
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1answer
28 views

Law of total probability explanation

What is the intuition behind the law of total probability? http://en.m.wikipedia.org/wiki/Law_of_total_probability
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1answer
24 views

Independent events and Kolmogorov

Suppose we have a probability space $(\Omega, \mathfrak{F}, P)$, and independent events $(E_n)_n$. Consider $$M_n = \sum_{k=1}^n I_{E_k}$$ Is it correct to say that by the Kolmogorov $01$ law $M_n$ ...
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2answers
25 views

Need a Probability Theory book that also focusses on Analysis

I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I ...
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8 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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28 views

Poisson Process Alterations

If we have a Poisson process of rate $\lambda$, do the following alterations still result in a Poisson process? 1) Deleting every alternate point 2) Inserting points at times $1, 2, 3, ...$ 3) ...
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2answers
38 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
4
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1answer
38 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
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1answer
43 views

Probability of a nonnegative submartingale converging to zero [on hold]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
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2answers
30 views

In Markov chains a limit distribution is invariant

Suppose we have a Markov chain $(X_n)_{n \geq 0}$ with state space $S$. Suppose that $(\pi_i)_{i \in S}$ is a limit distribution. Then is $(\pi_i)_{i \in S}$ an invariant distribution ? I know the ...
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0answers
29 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
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1answer
7 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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28 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a random subset of natural numbers (i.e. This is a probability space of sequence of natural numbers sometimes ...
3
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2answers
50 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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0answers
19 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
23 views

Dyadic expansion

I'm reading the appendix in Billingsley book "Probability and measures" and I can't understand the following. If $$\sum_{i=1}^n\frac{d_i(\omega)}{2^i} < \omega \leq ...
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1answer
15 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
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1answer
9 views

Smallest (sub-) Sigma algebra of a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
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25 views

Billingsley 2.5a) [on hold]

The field $\mathfrak{F}(\mathcal{D})$ generated by a class $\mathcal{D}$ of subsets of $N$ is defined as the intersection of all fields over $N$ containing $\mathcal{D}$. (a) Show that ...
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1answer
23 views

argument technique to prove convergence of random variable

I witness a lemma in my class note and I think the proof is not quite clear. Could anybody give me some ideas about argument technique to prove the lemma? The lemma 3 in the beginning of the text: ...
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1answer
17 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
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1answer
21 views

Why is $P(a < x < b) = P(a < x) - P(x < b)$?

Why is $P(a < x < b) = P(a < x) - P(x < b)$? This is an oversimplified version of a statistics problem I am doing, but I cannot remember why this is true. I know that this works and will ...
1
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1answer
25 views

Does the Strong Law of Large Numbers imply the following?

The Strong Law of Large Numbers in my Probability textbook is given as follows. Let $X_n$ be a sequence of identically distributed pairwise independent $\mathbb{R}$-valued random variables. Let $S_n$ ...
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1answer
35 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
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14 views

HOW TO PLOT DAG (DIRECTED ACYCLIC GRAPH) in BNT toolbox for matlab.

I have used markov chain monte carlo (MCMC) in BNT toolbox for matlab, from which i have got one output "sampled_graphs " which is cell array. Now how to plot DAG (Directed acyclic graph ) from ...