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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1answer
84 views

On unions of independent events

If the events $\{ E_{\alpha}, \alpha\in A\}$ are independent, then so are the events $\{F_\alpha,\alpha\in A\}$, where each $F_\alpha$ may be $E_\alpha$ or $E_\alpha^c$; also if $\{A_\beta, \beta\in B ...
0
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1answer
67 views

Expectation of function of stochast

I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
0
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1answer
138 views

Martingale and indicator

Exercise comes from "1000 exercices in probability" (12.9.6). Let $X_1, X_2, \dots$ be independent random variables with $X_n=\begin{cases} 1, & \text{with probability} & (2n)^{-1}, \\ 0, ...
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0answers
72 views

How does this violate probability theory?

Given: $X = Y^2 + Z^2$ (hence $E[X] = E[Y^2] + E[Z^2]$) $p(X = 1) = .52$, $p(X = 4) = .24$, $p(X = 16) = .24$ $p(Y = -1) = .5$, $p(Y = 3) = .5$ Question: Despite not being handed any information ...
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2answers
203 views

Uniform distribution on the n-sphere.

I have the next RV: $$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$ where $$X_i \tilde \ N(0,1)$$ It's a random vector, and I want to show that it has a uniform ...
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1answer
2k views

Joint distribution of multiple binomial distributions

In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them. The original file can be ...
2
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0answers
152 views

Area of a Random Polygon

The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
3
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3answers
4k views

Fisher information of a Binomial distribution

The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
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2answers
110 views

Law of large numbers?

Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$: If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
0
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1answer
103 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
0
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1answer
99 views

Approximation of a random variable by a sequence of simple random variables

It said in a probability book that any non-negative random variable $X$ can be approximated by a sequence of simple random variables (finite range) $X_1,X_2,\dots,X_n$ such that ...
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2answers
136 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
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1answer
85 views

Finding an expression for a multi variate joint CDF.

Let $X,Y$ and $Z$ be random variables with $X$ and $Y$ dependent, and $Z$ independent of both $X$ and $Y$. Let $f_{X},f_{Y},f_{Z}$ denote the density function's of $X,Y$ and $Z$ respectively and ...
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1answer
2k views

Finding an expression for the probability that one random variable is less than another, given a condition.

Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as: $$\mathbb{P}[X < Y] = ...
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4answers
294 views

Independent and uniformly distributed on $(\frac{1}{2},1]$

I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to ...
6
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2answers
111 views

$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$

If $\{X_n\}$ is a sequence of identically distributed r.v.'s with finite mean, then $$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$$ The inequality $$\frac{1}{n}E(\max_{1\le j\le n} |X_j|) ...
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3answers
132 views

A basic doubt on Lebesgue integration

Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate ...
2
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2answers
111 views

$P[X=Y]=0$ if $X,Y$ are i.i.d. with continuous c.d.f.

I am having lots of trouble proving the following statement: Let $X,Y$ be two real valued random variables on a probability space $(\Omega,\mathcal{F},P)$. These two variables are independent and ...
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1answer
637 views

Multivariate normal distribution density function

I was just reading the wikipedia article about Multivariate normal distribution: http://en.wikipedia.org/wiki/Multivariate_normal_distribution I use a little bit different notation. If $X_1,...,X_n$ ...
1
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1answer
163 views

A problem on almost sure convergence

Consider a sequence of random variables defined on the standard unit interval probability space : $$X_n = \begin{cases} 2^n & \text{when} \quad \frac{1}{2^n} \leq \omega \leq ...
2
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1answer
37 views

If $x^p P(|X|>x|)=o(1)$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$

If $p>0$ and $x^p P(|X|>x|)=o(1)$ as $x\to\infty$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$. It feels like the assumptions should lead to something like $\sum_n^\infty ...
2
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2answers
81 views

$ E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p$

The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory". If $p>1$, we have $$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$ ...
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2answers
96 views

Two random variable with the same variance and mean

Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely. My partial answer: By the definition of conditional expectation we have ...
2
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1answer
195 views

Exponential Levy process

We assume that the stochastic process L is a Levy process with the predictable characteristics triplet $(b,c,\nu)$. Which integrability conditions we should assume for the new stochastic process ...
4
votes
2answers
100 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
1
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1answer
91 views

Inequality between 2p-norm and p-norm for random variables

Recently I was studying something about random matrix theory, and class of sub-guassian / sub-exponential random variables is of interest. In the literature it gave an inequality as following: ...
1
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2answers
411 views

Moment generating function of a stochastic integral

Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then: $$ \mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
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1answer
97 views

A Measure For The Space of Probability Density Functions

Consider the space of all joint probability density functions of two variables. I want to know what the measure is of the portion of this space that is filled by uncorrelated joint pdfs relative to ...
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2answers
94 views

What exactly does this physically mean?

Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the ...
0
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1answer
53 views

Convergence of random variable to a negative constant

Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$ I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
4
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1answer
93 views

Measurability of an Indexed Product-Measure

If for any fixed $\omega_1$, $P_{\omega_1}$ is a probability measure and $Q_{\omega_1}$ is a stochastic kernel and both are measurable in $\omega_1$, is the indexed product measure ...
1
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1answer
105 views

probability of divisibility

Let S be the sum of k randomly selected integers between 1 and n. What is the probability of S being divisible q? Can this be expressed in a closed form? This is the generalization of one of the ...
1
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0answers
94 views

Haar system and martingale

Today our lecturer used martingale theory to show that the Haar system is a basis for $L_1[0,1]$ (we're operating on the probability space $([0,1],\mathcal{B},P)$, where $P$ is the Lebesgue measure). ...
4
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1answer
66 views

$P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$

If $E(X^2)=1$ and $E(|X|)\ge a >0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$. I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the ...
3
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0answers
213 views

History of odds making in sports betting

Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
0
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1answer
153 views

Conditional expectation is square-integrable

I am given the following definition: Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events ...
3
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5answers
180 views

Probability of sequence

The question is as follow: Let $(X_{n})_{n}$ be a sequence of Random variable that is independent with the probability $P(X_{n}=1)=1-P(X_{n}=0)=\frac{1}{n}$ Show that $P\left( \underset { ...
9
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2answers
543 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
2
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1answer
115 views

An application of Donsker's theorem.

Let ${X_i}$ be iid with $E[X]=0$ and $Var(X)=\sigma^2$ Let $S_0=0$ and $S_n=X_1+...+X_n$ for all $n \ge 1$. Show that $\lim_{n\rightarrow \infty}P(S_k>0 \space for \space k=n,n+1,\dots,2n)=1/4 $. ...
4
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1answer
55 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
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1answer
40 views

Probablistic approach

I have been reading about probabilistic approach in some problems in particular when we want to prove that something exists without explicitly constructing it. I really want to see more of this. Does ...
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1answer
245 views

Integrate over the uniform distribution on the simplex

Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int ...
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2answers
82 views

$\mathcal{A}\perp_\mathcal{G}\mathcal{B}\wedge\mathcal{H}\subseteq\mathcal{G}\implies\mathcal{A}\perp_\mathcal{H}\mathcal{B}$?

If $\mathcal{A}\perp_\mathcal{G}\mathcal{B}$ and $\mathcal{H}\subseteq\mathcal{G}$, is it the case that $\mathcal{A}\perp_\mathcal{H}\mathcal{B}$? Here $\mathcal{A}$, $\mathcal{B}$, $\mathcal{G}$ and ...
2
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0answers
71 views

Proving that $ \int \left| f-g \right|~d\mu = 2\int_{A_0} (f-g)~d\mu$

Given a (dominant) measure $\mu$, consider two probability measures $f~d\mu$ and $g~d\mu$ over $(\Omega, \mathcal F)$, I'd like to check the following reasoning for showing that $$ \int \left| f-g ...
1
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0answers
50 views

limit distribution of possion distribution

Assume Xn is possion with mean $\lambda_n$ and suppose that $\lambda_n\rightarrow\infty$ as $n\rightarrow \infty$. Then how to show Xn is AN$(\lambda_n,\lambda_n)$. I've tried to use characteristic ...
2
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0answers
509 views

Given the variance of a zero mean random variable X, what is the largest and smallest possible value for E[exp(-jX)]

I have a question that I am really eager to know the answer. It is as follows: Given the variance of a zero mean random variable $\mathbf{X}$ is $\sigma^2$, what is the largest and smallest possible ...
2
votes
1answer
140 views

Fun with Powerball history. What probability theories apply?

I wanted to play around with probability theories and see if I find any statistical inferences from the Powerball lottery history going back to 11/1/1997. What theories could I apply besides simple ...
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0answers
59 views

Question in ergodic theory

Again the source is http://www.math.ucla.edu/~biskup/275c.1.13s/PDFs/HW1.pdf this time I'm looking at #6 the part that is left as an open-ended question. If $f \in L^1$ and $\phi$ is a measure ...
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0answers
63 views

Homework questions in ergodic theory

Let $X_1, X_2, ...$ be iid. If $f: \mathbb{R}^\mathbb{N} \rightarrow \mathbb{R}$ is measurable wrt the product structure it's $L^1$ under the distribution measure induced by the $X_i$ then why is it ...
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0answers
60 views

Almost Surely Convergence

I need some help with computing the lim inf and lim sup of $ \frac 1n \sum_i X_n$ where the density of variable $X_n$ is absolute continuous, say, f(x) = exp(-x). I am interested in using the ...