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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2
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1answer
85 views

Is a function of two independent variables still independent?

Let a sigma-algebra $\mathcal{F}$ be given, and two random variables $X,Y$ independent of $\mathcal{F}$. Is it possible to conclude that for any measurable function $f$, then $f(X,Y)$ is still ...
2
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1answer
144 views

How many sets is in the $\sigma$- field generated by $n$ distinct sets

Given a set $\Omega$, let $A_1,\ldots, A_n$ be distinct subsets of $\Omega$. How many sets do you have in the $\sigma$-field generated by $\{A_1, A_2,\ldots, A_n\}$.
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2answers
181 views

Probability with people

Experience shows that 20% of the people reserving tables at a certain restaurant never show up. If the restaurant has 50 tables and takes 52 reservations, what is the probability that it will be able ...
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3answers
972 views

Conditional Probability Example

A drawer contains 8 different pairs of socks. If 6 socks are taken at random and without replacement, compute the probability that there is at least one matching pair among these 6 socks.
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1answer
45 views

Can the sum of two pregaussian random variables be zero?

Do there exist any two Sub-Gaussian random variables with variance 1 such that the sum of them is the $0$ random variable, whose value is $0$ with probability $1$?
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1answer
75 views

Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
1
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1answer
143 views

Local martingales

I want to be sure I understand the definition correctly, say as in here: http://en.wikipedia.org/wiki/Local_martingale I have grown accustomed to thinking of there being 3 classes of martingales. ...
3
votes
0answers
71 views

Question regard almost sure convergence

I need an example of an i.i.d. sequence of random variables $(X_i)$ such that $$ X_1\cdot\left(\frac{1}{n}\sum_{i=1}^nX_i\right)\not\to X_1\mathbb{E}(X_1) $$ almost surely as $n\to\infty$. ...
0
votes
1answer
54 views

a physical question about probability

assume that we have a cubic box which contains a large number of molecules. Therefore we know that the molecules move in different directions and hit the walls of the box . I read somewhere that with ...
2
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2answers
192 views

Ergodicity of a sequence of independent blocks

I am stuck with the problem given below, more precisely, with the part regarding ergodicity. I have a proof, also given in what follow, but it does not seem to be correct; well, at least, it does not ...
2
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0answers
340 views

Comparing two probability distributions

In my research I have to find two discrete probability distributions by solving two separate linear programs. The domain of optimization is the probability space of $m^n$ atomic events, where $n$ is ...
0
votes
2answers
94 views

question on poisson random measure

Assume $(S,\mathcal{S},\eta)$ is an arbitrary $\sigma$-finite measure space. Let $N:\mathcal S\rightarrow\{0,1,2,\ldots\}\cup\{\infty\}$ in a way such that $\{N(A):A\in\mathcal S\}$ are random ...
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2answers
53 views

Expected average of number of tiles empty

I have a grid of 3x4 and every tile on the grid has a corresponding chance of being empty. This chance is given. For example, consider the following grid: $$ \frac{14}{15} \frac{1}{3} \frac{8}{13} ...
1
vote
1answer
50 views

Show that, for a random walk $X$ on $\mathbb Z_2$, $X_\infty$ is independent of the past $\left\{ X_t \right\}_{0\le t<\infty}$

Define the *-finite time set $T=\{0,1,\ldots,n\}$, with $n \in {^*\mathbb N} \setminus \mathbb N$. Let $X=\left( X_t \right)_{t \in T}$ be a random walk on the cyclic group $\mathbb Z _ 2 = \mathbb Z ...
1
vote
1answer
279 views

Martingale with bounded increment.

In Rick Durrett's book "Probability: Theory and Examples", there is a theorem about martingales with bounded increment: Theorem 5.3.1. Let $X_1, X_2, \ldots$ be a martingale with $|X_{n+1} − X_n | ...
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2answers
2k views

Prove that the CDF of a random variable is always right-continuous

Let $X$ be a random variable with cumulative distribution function $F_X$. It is a known fact that this function $F_X$ is right-continuous. But I'm having some trouble to prove this result. Below I'm ...
1
vote
1answer
74 views

Atoms in a countable space

Let $(\Omega, \mathcal{F})$ be a measurable space where $\Omega$ is countable. I am trying to prove that there is some partition $\mathcal{P}$ of $\Omega$ such that the $\sigma$-algebra ...
3
votes
0answers
116 views

Upper semicontinuity of a probability measure

Let $m$ be an atomless probability measure on $\mathbb{R}^m$. Consider $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that for all $v \in \mathbb{R}^m$, $x \mapsto f(x,v)$ is ...
3
votes
1answer
191 views

Sigma Algebra Notation

Let $\mathcal{F}$ be a sigma algebra in $\Omega$ and let $A \subset \Omega$. I was reading a paper and I came across $A \vee \mathcal{F}$. What does this normally mean?
2
votes
1answer
138 views

Is the bijection of independent random variables, independent random variables?

I'm trying to prove the following: $$M_{X}(t)=\prod\limits_{i=1}^n M_{X_i}\left(\frac{t}{n}\right)$$ Where $X := \frac{1}{n} \sum\limits_{i=1}^n X_i $ and all the random variables are independent. ...
3
votes
1answer
271 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
0
votes
1answer
231 views

A gardener plants three maple trees, four oaks, and five birch trees in a row. …

The question and solution is taken from here. Question: A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally ...
3
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0answers
105 views

Mathematics Courses for an Economist

I am an Economist and I am interested in further developing my mathematical knowledge and skills. I would like to get your opinions on the topics that I should cover and which are also important for ...
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3answers
150 views

expected value of a $e^{-2|x|}$

Let $f(x)=e^{-2|x|}$ Find: $E(X)$ $E(|X|)$ $E(x')$ where $x'$ denotes the largest integer not greater than $x$. I'm stuck on this question and am confused about how to use the modulus sign. I ...
0
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0answers
161 views

Blackwell–Girshick equation

I came across Blackwell–Girshick equation days before,but I have found nothing about it. What is Blackwell–Girshick equation? Can you give some explaination or links?
2
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1answer
96 views

Learn about reproducing kernel Hilbert spaces?

Why are reproducing kernel Hilbert spaces an important topic to learn? What is possibly achievable with that theory that is not reachable with just standard Hilbert space theory?
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1answer
538 views

subadditivity proof

How do I prove: $P(\bigcup E_n) \le \sum_1^\infty P(E_n)$ I understand here, I need to expand the Union and because I do not have the disjoint sets assumption, the fallout will prove it. How do I ...
3
votes
1answer
28 views

Clarification regarding the statement of a lemma in theory of Martingales.

The following is taken from the book "Probability with Martingales" by David Williams Suppose that $T$ is a stopping time such that for some $N$ in $\mathbb{N}$ and some $\epsilon>0$, we have, ...
5
votes
1answer
122 views

Expected overlap

Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large ...
3
votes
1answer
144 views

Construct a sample space $\Omega$

Construct a sample space $\Omega$ and events $A_1,\ldots, A_n$ ($n\ge2$) such that $\operatorname{Pr}(A_i) = \frac12$ ($1 \le i \le n$), every $n-1$ of the $A_i$ are independent, but the $n$ events ...
1
vote
2answers
530 views

A tighter bound than Markov Inequality

Is there any way to find a tighter bound than Markov inequality? For example, I know that if always $X>b>0$, we can define $Y=X-b$ and have a tighter bound. Is there any similar solution that ...
1
vote
1answer
112 views

Probability using volumes wedge

Suppose that a point $(X, Y, Z)$ is chosen uniformly at random from the wedge $f(x ,y,z)$ belongs to $\mathbb{R}^3: 0 \leq x, y \leq 1, \textrm{and}\, 0 \leq z \leq x$. Compute the probability $(a ...
3
votes
1answer
235 views

Non-convergence of Cauchy Random Variables

Suppose $X_1,X_2,\ldots$ is a sequence of Cauchy random variables with density $$f(x)=\frac{1}{\pi(1+x^2)}, \hspace{3mm}x\in \mathbb{R}$$ and let $S_n=X_1+\ldots+X_n$. It's easy to show that ...
2
votes
2answers
4k views

Law of Iterated Expectations example

I have a question that I hope can be shown using the LIE. There is a urn which contains 3 marbles (2 white and 1 black). The person who gets to pick the black marble gets to win $100 whereas the ...
2
votes
2answers
112 views

When does convergence in $L^p$ imply convergence of the p-th moment?

Suppose $X_n$ is a sequence of random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. When does convergence in $L^p$,i.e. $$\mathbb{E}[\vert X_n - X \vert ^p]\rightarrow 0,$$ ...
5
votes
1answer
117 views

Solutions of a cubic diophantine equation in $\mathbb{Z}/p\mathbb{Z}$

Suppose $p\in\mathbb{Z}$ is prime and $p\equiv 1\pmod{3}$. Is there an estimate of the number of solutions of $x^3+y^3=z^3$ in $\mathbb{Z}/p\mathbb{Z}$, preferably using elementary number theory and ...
1
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0answers
187 views

Intuition about absolute Continuity/Singularity of measures

Let $\mu$, $\nu$ be two measures, $X= d\nu/d\mu$ their Radon-Nikodym derivative (in general a random variable). I want to gain an intuition about the following statements: $$\int X\; d\mu = 1 ...
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2answers
257 views

The probability that two vectors are linearly independent.

Given the set $R^{n}$, what is the probability that 2 vectors drawn are linearly independent? I know the answer is almost 1 but I am not able to reason. Could someone please help me out?
3
votes
1answer
165 views

Criterion for independency of random variables

I saw in some notes the following "criterion" for independency of two random variables. Let $X$ and $Y$ be real-valued random variables defined on the same space. $X$ and $Y$ are statistically ...
1
vote
1answer
182 views

Distribution of sum of mixed random variables

Suppose there are $n$ i.i.d mixed random variables $U_1,U_2,\cdots,U_n$. Each has a mass of probability $e^{-\tau}$ at $0$ and a pdf $$f(u)=\left\{\begin{matrix} e^{-u}, & 0<u\leq \tau\\ 0, ...
3
votes
2answers
129 views

Law of large numbers with weights

Let $X_1 , X_2 , ...$ be an iid sequence with mean 0 and finite variance. Let $(a_n)$ be a sequence of non-random real numbers. Under what conditions on $(a_n)$ do the weighted means $\frac{1}{n} ...
0
votes
3answers
112 views

solutions poker texas hold'em

Is there any equation that characterizes the poker game in terms of variables such as the strength of the hand, the amount of betting money in the pot, etc? Is there any solution that says what the ...
3
votes
1answer
95 views

Almost surely statement in Williams book.

In Probability with Martingales (Williams) I came across the following proposition and then they give the following contradictory example Could someone please explain how it can be so? Also why is ...
4
votes
1answer
147 views

An intuitive solution to this problem (Using probability tree)

A group of boys has been lost several days in the dessert. This group has a phone to make phone calls. After a long way walk, they believe that the current area is suitable for phone calls; even ...
5
votes
1answer
101 views

A maximal Hoeffding's inequality?

Let $X_1, \cdots, X_n$ be real-valued independent random variables satisfying $|X_k|\le 1$ and $\mathbb EX_k=0$. Hoeffding's inequality tells us that for any $k=1,\cdots, n$ and $t>0$, $$\mathbb ...
1
vote
1answer
78 views

Is there a basic solution to this probability problem?

There are 16 army squads going to protect 8 different cities (labeled from A to H). Due to weather conditions, it's impossible the communication between the squadrons. Each squadron must deicide which ...
3
votes
3answers
244 views

Different of mapsto and right arrow

Could someone please explain to me what is the difference in the two arrows$$\rightarrow$$ and $$\mapsto$$ For example in Probability wih Martingales (Willams) Thank you.
3
votes
2answers
103 views

Monotone convergence example

In the first chapter of Probability wih Martingales (Willams) I came across the following example. Book says it's wrong, I don't understand what is wrong in that. Could somebody please explain why ...
1
vote
1answer
90 views

General questions about Normal Distribution characteristics

I am very weak in understanding what my lecturer says because of many gaps in what I know. The Density Function of the Normal with parameters $(\mu,\sigma^2)$ is ...
1
vote
2answers
153 views

Ugly Subsets: Weirdness Within the Axioms of Probability Theory

I'm watching this video now, and at $36:53$ John Tsitsiklis mentions that for some sets there is no way to assign probabilities to events which occur in them. I'm wondering what sets he is talking ...