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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Sub sigma-algebra Example

I'm looking at the sub $\sigma$-algebra example on Wikipedia, and I don't understand the notation that is used. The example defines the $\sigma$-algebra $G_n = \{ A \times \{H,T\}^{\infty}\ : A ...
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17 views

Gaussian Free Field and Branching Random walk

I was told that there is a duality between gaussian free field in 2-D and branching random walk. Could anybody please give a brief explanation how both probability theory are dual to each other? ...
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1answer
23 views

Where do the forumlas for expectation and variance for geometric and Poisson distributions come from?

Okay so I have been given a list of 4 distributions and their respective mean(expected) and variance. I can see where the Bernoulli and Binomial ones come from using the definition of expectation and ...
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2answers
106 views

Bounding profits of gambler by Azuma Inequality

A gambler plays the following game: In each round, he can pay any $0 < p < 1$ dollars, and win 1 dollar with probability p (independently). Show that the probability that the gambler's net ...
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69 views

An example of switching expectation and infimum

Suppose $(X_t : t \geq 0)$ is a real valued stochastic process. In my case, it takes values in $[0,1]$, starts at $1$, and is decreasing. (It is also Markov). I have calculated $\mathbb{E} (X_t) =: ...
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1answer
41 views

Find a sequence of events $A_n$ for which all three inequalities…

Let $(\Omega,F,P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Find a sequence of events $\{A_n\}$ for which these inequalities hold: (i) For each ...
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2answers
138 views

Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
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1answer
28 views

Finding measure given constant margins

Suppose $g:[0,1]^2\to R$ and $g$ can have finitely many discontinuities. $F$ is continuous and atomless c.d.f on $[0,1]$ $$\int_{[0,1]} g(x,y)dF(y)=1/2, \forall x$$ $$\int_{[0,1]} g(x,y)dF(x)=1/2, ...
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39 views

A question about weak convergence of random variables

I am reading my lecture notes and our definition of weak convergence or random variables is: First another definition: A sequence $\mu_n$ of probability measures on $\mathbb R$ converges weakly to a ...
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1answer
68 views

Continuity of the Second Moment of a Continuous Stochastic Process

Given a continuous stochastic process with respect to time with finite variance at a given $t$. Does it necessarily imply the second moment and variance are continuous in time?
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41 views

recurrence/transience on random walk

Let $X_n$ be a markov chain, $p>\frac{1}{2}$ and $E=\{0,1,2,...\}$ its state space. Let $\Pi$ be its transition matrix with $\Pi(0,0)=p$, $\Pi(i+1,i)=p$, $\Pi(i,i+1)=1-p$ , $i\ge0$. ...
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1answer
31 views

Convergence of random variables: struggle with a proof

I am trying to understand a proof of the following theorem: $X_n$ is a sequence of random variables. $X_n \to X$ in probability $\implies$ that each sub-sequence of $X_n$ has a sub-sequence which ...
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1answer
31 views

Fingerprinting and randomized algorithms

My question is regarding the notes pages 1-2 specifically http://www.cs.berkeley.edu/~sinclair/cs271/n3.pdf I understand everything up to the point near the top of the second page where it says ...
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1answer
30 views

Card shuffling transition matrix

a short understanding question. Consider a pile of $n$ cards. At every step we choose randomly 2 cards and transpose them. Now $X_n$ should be a Markov chain which describes the order of the pile at ...
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1answer
82 views

Determining a transition probability matrix

If I have that $X_n$ is a two-state Markov chain whose transition probability matrix is: $P = \left( \begin{smallmatrix} \alpha & 1-\alpha\\ 1-\beta & \beta \\\end{smallmatrix} \right)$ ...
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1answer
30 views

tail inequality for expectations

I would like to upper bound the expectation $$ \mathbb{E}[X \, \textbf{1}\{X > t\}], $$ where $\textbf{I}\{p\}$ evaluates to $1$ if $p$ is true, $0$ otherwise, and $X$ is some non-negative random ...
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2answers
91 views

Approximation of conditional expectation

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. Let $\mathcal{A}$ be a complete sub-$\sigma$-algebra of $\mathcal{F}$. For the moment assume that $X$ is a random variable with ...
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48 views

Proof that limit exists in $L^2$ sence

Proof that exists $L^2$ limit $$ \lim_{\varepsilon\downarrow 0} L(t,\varepsilon)=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon}\int_0^t\mathbf{1}\left(W_s\in(-\varepsilon,\varepsilon)\right)ds, ...
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1answer
51 views

Joint Probability and Intersection Probability

Given two independent events A and B: $P(A \cap B)= P(A)*P(B)$ but then I saw somewhere that: $P(A \cap B)= P(A)*P(B)= P(A|B)*P(B) = P(B|A)*P(A)$ where for example $A$ is $X=x$ and $B$ is $Y=y$ ...
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1answer
33 views

Elementary event in event space

I encountered a very basic question of probability. Consider the sample space Ω = {a,b,c,d} and assume that the only elementary events in the Event space F defined on Ω are {a} and {b}. Explicitly ...
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1answer
38 views

How to find $\mathbb P(XY<\frac{1}{2})$ and $\mathbb P(Y< X^2)$ without convolution?

Let $X$ and $Y$ be uniformly distributed on $[0,1]$ and independent random variables. Find $$\mathbb P\left(XY<\frac{1}{2}\right) \text{ and } \mathbb P\left(Y< X^2 \right).$$ Tip: one can do ...
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1answer
28 views

Independent identical distributions and conditions for equality

Suppose $X,Y,Z$ are independent identical distribution, taking values in a finite set $x_1,...,x_n$. Is the following true? $$Pr\{X=Y: Y\neq Z\}\leq Pr\{X=Y\}$$ What do think about the condition for ...
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2answers
42 views

Proving certain aspects of Entropy

I am trying to prove three properties of entropy. $1)$ $H(X|Y,Z)\le H(X|Y)$ $2)$ $H(X|Y,Z)\le H(X,Y)$ $3)$ $H(X,Y,Z)+H(Y)\le H(X,Y)+H(Y,Z)$ I have proved the third one, but it is based on part 1. ...
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1answer
41 views

Series of moments of random variables

I am interested in the convergence of the series $$ \sum_{n=1}^\infty\frac1n\operatorname E[|X|^pI_{\{|X|>b_n\}}], $$ where $X$ is a random variable with $\operatorname E|X|^p<\infty$ for some ...
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1answer
24 views

Describe the push forward of the borel probability measure on $\mathbb{R}$ under its Cumulative distribution function

Will the push forward be the borel probability measure on [0,1]. If so, how to show this? Kindly help
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53 views

Prove that a process has independent increments

Let $\tau_1,\tau_2,\dots$ be a sequence of i.i.d. random variables such that $\tau_i \sim Exp(\lambda)$ on $(\Omega,\mathcal E,\mathbb P)$. Then define $T_i=\sum_{k=1}^i \tau_k$, so we know how each ...
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0answers
46 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
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0answers
41 views

If $E(|X|\log|X|)<\infty$ then is $E\left[\frac{|S_n|}{n}\ \log\left(\frac{|S_n|}{n}\right)\right]<\infty$?

I am trying to finish a homework problem in my probability class. I think I am at the end of my problem if I can show that $$E(|X|\log|X|)<\infty$$ implies that $$E\left[\frac{|S_n|}{n}\ ...
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0answers
20 views

Proving the sample variance has a chi squared critical value

Let $X_1, . . . , X_n$ be independent normal observations with means $µ = 0$ and variances $σ^2$. For testing the null hypothesis $H_0 : σ^2 = 1$ versus the alternative $H_a : σ^2 > 1$ show that ...
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2answers
51 views

Continuity of $\mu \mapsto \mu(E)$ for $\mu$ probability measure and $E$ Borel subset

Let $X$ be a topological space endowed with the Borel sigma-algebra, let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, endowed with the weak* topology. Fix $E$ Borel subset of ...
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1answer
8 views

If Y and X are ind. binomial RV with parameters (n, p) and (a, b) respectively, then (Y/n) - (X/a) is approximately distributed. Find V(Y/n - X/a).

I tried to find E(Y/n - X/a) and said it was E(Y/n) - E(X/a)= p - b. But then I got stuck finding the variance, I wasn't sure if it needed to be done with moment generating functions or if the Central ...
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1answer
66 views

Show that if the sum of an diverges, no discrete probability space can contain independent events

Suppose that $0\leq p_n\leq 1$, and put $a_n= \min \{p_n, 1-p_n\}$. Show that if $\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1, A_2, \ldots$ such that ...
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2answers
113 views

Find three events that are dependent but pairwise independent

Let $(\Omega, \mathcal F, P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Q. Give an example of three dependent events with probabilities strictly ...
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1answer
57 views

Prove that the following is a field. [closed]

Let $\mathcal{F}_1,\mathcal{F}_2,...$ be classes of sets in a common space $\Omega$. (a) Suppose that $\mathcal{F}_n$ are fields and that $\mathcal{F}_n\subset\mathcal{F}_{n+1}$. Show that ...
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1answer
35 views

Does this argument suffice to show a “record” occurs at time n with probability 1/n?

I think it does, but, in addition to checking for correctness, I'd like to know what other argument we might use. Let $X_1, X_2,...X_n$ be be a sequence of independent identically distributed ...
3
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1answer
56 views

$X_n \stackrel{d}{\to}X$, $Y_n \stackrel{d}{\to} c \implies X_n+Y_n \stackrel{d}{\to} X+c$

Let $X_n\Rightarrow X$ and $Y_n\Rightarrow c$. Show that $X_n+Y_n\Rightarrow X+c$. Prove: There exists sequences of random variables $(X^{(*)}_n)$ and $(Y^{(*)}_n)$ such that $(X^{(*)}_n)$ and ...
3
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1answer
34 views

Fatou for weak convergence

I want to do exercise 3.2.4 from Rick Durett, Probability: Theory and Examples page 86. $$\text{Let } g\geq0 \text{ be continuous. If }X_n \Rightarrow X_{\infty} \text{ then } \liminf_{n\rightarrow ...
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2answers
66 views

Expected first return time of Markov Chain

Given the following Markov Chain: $$M = \left( \begin{array}{cccccc} \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0 & 0 ...
3
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1answer
41 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
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1answer
44 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
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2answers
100 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
2
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1answer
30 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
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15 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
2
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2answers
99 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...
2
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1answer
31 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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20 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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19 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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1answer
102 views

Repeated coin flips probability

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated N times. We represent these in a table with $N$ rows and $L$ columns with order. So a column is defined at the position ...
3
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1answer
65 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
91 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? ...