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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Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
5
votes
1answer
206 views

Estimate variance given a sample of size one (7th Kolmogorov Student Olympiad)

This is problem 10 of the seventh Kolmogorov Student Olympiad in Probability Theory as translated by Jonathan Christensen in this thread. Given a sample of size one from the random variable $\xi ...
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1answer
785 views

Another question on almost sure and convergence in probability

Convergence in probability implies convergence on a subsequence almost surely. But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we ...
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1answer
37 views

Suppose $(X,Y) \overset{d}{=} (X,Z)$ and $Y$ is $Z$-measurable. Prove $X$ and $Z$ conditionally independent given $Y$.

This is a question out of Kallenberg's Foundations of Modern Probability. Suppose $(X,Y) \overset{d}{=} (X,Z)$ and $Y$ is $Z$-measurable. Prove $X$ and $Z$ conditionally independent given $Y$. I can ...
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1answer
47 views

Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathcal{F}$ a sub-$\sigma$-algebra. Let $\mathcal{F}$ be trivial, i.e. $\forall A\in\mathcal{F}: ...
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1answer
118 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 ...
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99 views

Measurability problem of sample distribution function of a contiuum of independent random variable

Let $I = [0,1]$ be the index set of a contiuum of i.i.d random variables. For each $t \in I$, the sample space of $X_t$ is $\Bbb R$ equipped with Borel $\sigma$-algebra and Borel probability measure. ...
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231 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
4
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1answer
103 views

Is it true that $E[|Y|\mid{\mathcal G}]\leq |Y|$ almost surely?

This is an exercise of conditional expectations: Let $Y$ be an integrable random variable on the space $(\Omega,{\mathcal A},{\bf P})$ and $\mathcal{G}$ be a sub $\sigma$-algebra of $\mathcal{A}$. ...
4
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1answer
106 views

Proof: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$

As the title states, the problem at hand is proving the following: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$ Attempt/thoughts on a solution I am guessing this is an ...
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157 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
4
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1answer
334 views

Conditional expectation: $E[A \mid B] = B$ and $E[B \mid A] = A$ implies $A = B$

Given two $L^1$ random variables $A$, $B$ satisfying $E[A \mid B] = B$, $E[B \mid A] = A$. The claim is that $A = B$. So is this easy? I don't really see it.. If $A$ and $B$ are in $L^2$ it's clear, ...
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4answers
440 views

The Expectation and the Variance of the runs

folks! I have the following problem: Suppose you have a coin that has chance p of landing heads. Suppose you flip the coin N times and let X denote the number of "head runs" in N flips. A "head run" ...
3
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1answer
82 views

Functions and convergence in law

Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ ...
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51 views

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers ...
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1answer
123 views

definition of “weak convergence in $L^1$”

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set ...
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3answers
365 views

Is a square-integrable continuous local martingale a true martingale? [duplicate]

I am wondering, if the following is true without any other assumptions and if so, how to prove it: Let $(M_t)_{t \geq 0}$ be a continuous local martingale on a filtered probability space $(\Omega, ...
3
votes
1answer
115 views

Understanding the definition of a random variable

I'm working through a math stats book on my own (I've always wanted to learn it), but I'm getting confused about the definition of a random variable. The book says that a random variable is a ...
3
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1answer
131 views

Infinite divisibility of random variable vs. distribution

The distribution of any infinitely divisible random variable is itself infinitely divisible. But this link says the converse is not always true. Can you explain?
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1answer
222 views

Modulo of sum of random variables

Let $X_i$ be i.i.d. random variables on $\{1,\dots,d\},d\in \mathbb N$. What is the best way to analytically treat $$\left(\sum_{k=1}^{n} X_k \right ) \mod d$$ Is there a general recipe to get rid ...
3
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1answer
382 views

$K$ consecutive heads with a biased coin?

You toss a coin repeatedly and independently. The probability to get a head is $p$, tail is $1-p$. Let $A_k$ be the following event: $k$ or more consecutive heads occur amongst the tosses numbered ...
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144 views

Every uncountable Polish Space has a copy of $\{0, 1\}^\mathbb{N}$

I am having trouble verifying corollary 7.8 on p. 6 in this document http://www.math.ucla.edu/~biskup/275b.1.13w/PDFs/Standard-Borel-Spaces.pdf My troubles are with the definition of the "tree" ...
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0answers
105 views

Failure criteria for a collection of independent evolving discrete random variables

I have built a computer model that contains a collection of independent discrete random variables. They each have values of between $0$ and $k$ where $k$ is between $4$ and $15$ and is constant for ...
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1answer
254 views

Random walk in the plane

A particles moves in $\mathbb{R^2}$ started at the origin. At each stage $i (i = 1, 2, ...)$, the particles would move, independently of all the stages before, one of the four directions North, East, ...
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2answers
1k views

Independence of sample mean and variance through covariance

I have seen the text book derivation where the independence is established through factoring the joint distribution. But has anyone tried to prove that covariance is zero?. Let $Z_{i}$ come from a ...
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2answers
317 views

What is a canonical version of conditional expectation?

In David Williams's Probability with Martingales, there is a remark regarding conditional expectation of a random variable conditional on a $\sigma$-algebra: The 'a.s.' ambiguity in the definition ...
3
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2answers
411 views

Convergence of variance and mean of weighted sums of random variables

Let $c_n$ be a sequence of positive real numbers such that $$\sum^{\infty}_{n=1}{c_n} = \infty, \qquad \sum^{\infty}_{n=1}{c_n^2} < \infty.$$ Let $X_n$ be a family of i.i.d random variables with ...
3
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1answer
243 views

Strong law of large numbers for uncorrelated $L^2$ random variables

I'm reading Probability Theory, and ran into the following exercise: Prove that if $X_n \in L^2$ are uncorrelated and identically distributed random variables, then $$\frac{1}{n} \sum_{k=1}^n X_k \to ...
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589 views

Characteristic Functions and motivations

I've recently studied characteristic functions in my probability course and I can't get why we define it to be the Fourier transform of the distribution (if the random variabile is continuous). I ...
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3answers
326 views

discrete version of Laplacian

Suppose $P(x,y)$ gives transition probabilities of a random walk. I've seen $Pf=f$ being called the discrete version of Laplace's equation. In what sense are they analogous?
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1answer
52 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
2
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1answer
65 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose density function $ f $ is given by $$ f(y) = \begin{cases} ...
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1answer
69 views

Why the two probability results are the same?

Suppose there are 3 red balls and 2 white balls in a bag. We want to pick out 2 balls without replacement. What's the probability of the 1st and 2nd balls are both red? Solution 1: Use the ...
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2answers
80 views

Infinite Inclusion and Exclusion in Probability

Is there some way of generalizing the principle of inclusion and exclusion for infinite unions in the context of probability? In particular, I would like to say that $P(\bigcup_{n=1}^{\infty}A_n) = ...
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2answers
552 views

Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = ...
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2answers
77 views

What can be said about $E[1_A\mid\mathcal F]$?

It is known that $E[X\mid 1_A]$ is of particularly nice form. What can be said about the form of $E[1_A\mid\mathcal{F}]$ for "general" $\mathcal{F}$? Is it true that $E[1_A\mid\mathcal{F}]=1_B$ for ...
2
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1answer
174 views

sigma algebra problem [duplicate]

Let $f$ be any function from $\Omega$ to $X$ and $\mathcal{C}$ an arbitrary nonempty collection of subsets of $X$. Show $$\sigma(f^{-1}(\mathcal{C}))=f^{-1}(\sigma(\mathcal{C}))$$ I already know how ...
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0answers
71 views

Allowed probabilities under frequentism

Am I right to assume that under the frequentist interpretation of probability,* the set of allowed probabilities isn't $$\left[0,1\right],$$ but rather is ...
2
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1answer
101 views

Mathematical expectation is inside convex hull of support

Let $\xi$ be a random variable supported in some set $A \in \mathbb{R}^n$: $\xi \in A$ a.e. How to show that $\mathsf E \xi \in \mathop{\mathrm{conv}}{A}$? Let $s(x)$ be a support function of set ...
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1answer
156 views

How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?

Consider $$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$ where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
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1answer
538 views

The law of absolute value of a standard Brownian motion

How can we easily compute $\mathbb{E} [ \left|W_t\right|]$, where $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?
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1answer
156 views

A problem of regular distribution

This is a exercise of Shiryaev's Probability on page 233: Suppose that the random elements $(X, Y)$ are such that there is a regular distribution $P_x(B)=P(Y\in B\mid X=x)$. Show that if ...
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1answer
89 views

Probability that a sequence of random variables converges to 0 or 1

Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and $$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) ...
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1answer
790 views

Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
2
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1answer
105 views

Finiteness of expected values for independent random variables

Show that if the random variables X and Y are independent and for some $p > 0$: $E(|X+Y|^p)<\infty$, then $E(|X|^p)<\infty$ and $E(|Y|^p)<\infty$. I'd share thoughts or working out, but ...
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1answer
246 views

Dynkin's Theorem, and probability measure approximations

I am working my way through Resnick's A Probability Path, and looking at Exercise 2.5 I am a bit stuck on the application of Dynkin here. The question states: Problem: Let P be a probability measure ...
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2answers
213 views

Convergence of $\frac{a_n}{n}$ where $a_0=1$ and $a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$

Given $a_0=1$ and:$$a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$$Find convergence or divergence of $\frac{a_n}{n}$. Such a weird problem; I don't know how to attack it. I'm also fairly ...
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1answer
3k views

Martingale that is not a Markov process

1. On the internet, it is suggested that $$ X_t=\left(\int_0^t X_s \;ds\right)\;dW_{t} $$ is a martingale, but not a Markov process. I understand that the process $$ I_t(C)=\int_0^t C_s \; dW_s$$ ...
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2answers
771 views

Absolute continuity of a distribution function

This appeared on an exam I took. $Z \sim \text{Uniform}[0, 2\pi]$, and $X = \cos Z$ and $Y = \sin Z$. Let $F_{XY}$ denote the joint distribution function of $X$ and $Y$. Calculate ...
2
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3answers
534 views

How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's

EDIT: new formulation of the question (old version below). In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...