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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54 views

The expected value of the smallest number in sample $S$ is:

We are given a set $X = \{x_1, …. x_n\}$ where $x_i = 2^i$. A sample $S ⊆ X$ is drawn by selecting each $x_i$ independently with probability $p_1 = \frac{1}{2}$. The expected value of the smallest ...
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42 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
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47 views

Condicional Expectation when $\mathbb{E}[X] = \infty$.

Let $(\Omega, \textit{F}_0, \mathbb{P})$ and $\textit{F} \subset \textit{F}_0$. Suppose $X \geq 0$ and $\mathbb{E}[X] = \infty$. Then there is a unique $Y \textit{F}$-measurable with $0 \leq Y \leq ...
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41 views

If $dQ=\Lambda\, dP$ then $E^Q\left[ X \mid \mathcal{G}\right]=E\left[ X \Lambda\mid \mathcal{G}\right]/E\left[ \Lambda \mid\mathcal{G}\right] $

The statement: Take two probability measures $\mathbb{P}$ and $\mathbb{Q}$ on $(\Omega, \mathcal{F})$, such that $\mathbb{Q}\ll\mathbb{P}$ with $$d\mathbb{Q}=\Lambda d \mathbb{P}.$$ Let ...
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31 views
+50

Convergence of moments of a sequence of random variables

I encountered this problem in my study of time series. It seemed trivial at first but I don't see the finishing move to complete the proof. The problem is as follows. Let $(X_n)_n$ be a sequence of ...
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18 views

Reshuffling the order statistic of uniform at midpoint

Let $U_1,\dots,U_n$ be i.i.d. drawn from the uniform distribution on $[0,1]$ and let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ be their order statistics. Assume that $n$ is even and that we take ...
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32 views

Probability Models (Poisson Process)

There are two types of claims that are made to an insurance company. Let $\ N_{i}(t)$ denote the number of type i claims made by time t, and suppose that $\{N_{1}(t): t \ge 0\}$ and $\{N_{2}(t): t ...
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16 views

Possibility of analogy of the Bayes theorem for expectations

I recently found a scenario where I wanted to find the relation of $ E[X|Y] $ and $ E[Y|X] $ for $ X,Y $ two random variables. For probabilities and densities we have the Bayes theorem which is well ...
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25 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
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16 views

“Local” functional central limit theorem for the empirical distribution function

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[X^2]<\infty$. Denote by $F_X(t) := \mathbb P(X\leq t)$ their common distribution function. ...
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33 views

Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0 < P(B) < 1$

Let $(\Omega, \mathbb{F}, P)$ be a probability space. Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0$ $<$ $P(B)$ $<$ $1$. My ...
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22 views

For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
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24 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
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24 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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30 views

Generalizing lemma 1 in Tao's notes on operator norms of random matrices

My question concerns the proof of Lemma 1 in this blog post of Terence Tao. In the first paragraph, he says: Applying standard concentration of measure results (e.g. Exercise 4, Exercise 5, or ...
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13 views

Tight upper bound for expectation of function of a positive and bounded random variable

This problem popped up in my research. Let $X$ be a positive and bounded ($X\in (0,B) \ a.s.$)random variable with degrees of freedom $d$ and noncentrality parameter $\lambda$. My goal is to find ...
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17 views

Expectation of the maximum of random variables

I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$, where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent. I've been trying to figure this out by doing, $E(\max \{ a-X, ...
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22 views

What happens when we pick a Random sample?

Let $(\Omega, \mathcal{F} , \mathbb{P})$ be a probability space and $X:\Omega \to \mathbb{R}$ be a random variable. When we simulate or pick a random sample of size $n$ from $X$, are we picking ...
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19 views

Conditional expectation of a hitting time of a Brownian motion and Laplace transform

I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let $\mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}$. For $a < 0 < b$ ...
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25 views

Finding a distribution whose KL Divergence from a given distribution is a constant $\alpha$

Consider P as a multinomial distribution over k variables. I would like to find a distribution Q, also a multinomial distribution over k variables such that KL Divergence between Q from P is a ...
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43 views

About directly finding the PDF vs finding the PDF from a CDF

Lets consider a simple question : Say $X$ and $Y$ are two random variables which are sampling uniformly from $[0,1]$. We want to compute the PDF of $XY$. The typical approach seems to be to compute, ...
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52 views

Find the probability mass function of a poisson distribution

A random sample $X_1, X_2, . . . , X_5$ is taken from a Poisson distribution with parameter λ for some λ > 0. Find the joint probability mass function in as simplified a form as possible for ($X_1, ...
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17 views

The definition of completion of one measure with respect to a family

The following is taken from the book of Revuz and Yor more or less verbatim. If $(E,\mathcal E)$ is a measure space carrying probability measure $\mu$, the completion $\mathcal E^\mu$ is the ...
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60 views

Pos properties.

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of ...
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41 views

Sample space of infinite coin tosses experiment

In one of the online statistics/probability web sites http://www.statlect.com/asymptotic-theory/mean-square-convergencee, the following definition is given for the mean - square convergence of the ...
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23 views

Poisson process deviations

How can one prove the following inequalities for a standard Poisson process $\mathbf{N}(t)$ ? $\mathbb{P}\bigg[\bigg|\frac{\mathbf{N}(\lambda)}{\lambda}-1 \bigg| > \varepsilon\bigg] \leq ...
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45 views

Recommendations: Any Good books to study Path-Integration from 0 again?

I was researching and talking with some friends about I want to start from zero studying path integral, this question, and they recommended I start by studying "Quantum Mechanics and Path Integrals". ...
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32 views

Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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19 views

Stopped process of maximum stopping times

Suppose $X$ is an adapted process and $\tau_1, \ldots , \tau_k$ are stopping times such that $X^{\tau_1}, \ldots , X^{\tau_k}$ are all martingales. I want to show that then $X^{\tau_1 \vee \ldots \vee ...
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29 views

Find distribution of a random variable sequence

Let $X_n$ be the sequence of random variables which have their values from $(0, n]$ for $n > 0$. The cumulative distributive function of $X_n$ is $F_{X_n}(x) = 1 - (1 - x/n)^n$ for $0 < x \leq ...
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45 views

Notion of conditional weak convergence

I am looking for references or lecture notes which define the notion of conditional weak convergence of a sequence of random variables. In the case of (usual) weak convergence, we say that a sequence ...
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17 views

Asymptotic property of joint distribution

Random variables $X_1,X_2,\cdots,X_n$ have joint probability distribution $\mathbb{Q}_{X_1,X_2,\cdots,X_n}(x_1,x_2,\cdots,x_n)$. Assume another probability function $\mathbb{P}_Z(z)$. What does ...
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25 views

Existence of $L^2$ random variable conditional expectation.

Assume $X \in L^2(\Omega, \mathscr{F}, \mathbb{P}).$ Let $\mathscr{G} \subset \mathscr{F}$. I want to deduce how to define $$\mathbb{E}(X|\mathscr{G}).$$ Set $$s := \sup_{\mathscr{H}\subset ...
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44 views

Coarsest filtration

Let $(\Omega,\mathscr F, \mathsf P)$ be some probaility space, $T = [0,+\infty)$ and $\eta = (\eta_t)_{t\in T}$ be some real-valued stochastic process. I say that a stochastic process $\xi$ is good if ...
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26 views

Sized Biased Picking Distribution

I am having trouble understanding the following proof on sized biased picking. We have the following situation: Let $ X_1, \cdots , X_n $ be i.i.d. and positive, and $S_i = X_1 + \cdots + X_i$ for $ ...
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31 views

The limit of Kullback-Leibler divergence

Let $X_1, X_2, \dots, X_n$ -- i.i.d. sample. Suppose we have two models: $$M_0: X_1, X_2, \dots, X_n \sim N(0, 1) \\ M_1: X_1, X_2, \dots, X_n \sim N(\theta, 1)$$ So we have two hypothesis: $H_0: ...
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30 views

Variation on the classic ABRACADABRA problem

Suppose letters are chosen randomly from the alphabet, one at a time. It is well known that the expected time until ABRACADABRA is spelled out is $26^{11}+26^{4}+26$. The proof uses discrete time ...
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23 views

Distribution that describes # of memoryless events in an interval but the mean is not constant.

Context: I'm taking a stochastic processes class right now and we got a bit into queueing theory. In all the queue's we've considered, arrivals follow a poisson process. This seems unrealistic in some ...
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33 views

Are Ito Integrals adapted to the Brownian Motion Filtration

Give a probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_t, P)$, we could define a 1-dim Brownian motion $W_t$ adapted to $\{\mathcal{F}_t\}_t$ with its own filtration $\mathcal{F}_t^W$. For ...
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28 views

Convergence in distribution implies convergence of sequence

Let $(X_n)$ be a sequence of random variables. Let $(a_n), (b_n)$ be sequences of real number. Assume that there exists random variables $X, Y$ such that $$X_n \rightarrow^{d} X, a_nX_n + b_n ...
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If $B_t - B_s, \ 0\leq s < t,$ is normally distributed, there are constants $C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n$

I am working on the following problem: Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive ...
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27 views

Generating continuous random variables from a set of Bernoullis

Given a set of $Bernoulli(p_i)$ variables with each having its own arbitrary $p_i$, is there an efficient way to generate continuous random variables sampled from an arbitrary distriubution? To ...
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16 views

Fourier transform of a probability measure, and fourier transform of density

I have defined, for a probability measure $\eta$ we have the fourier transform as $\hat{\eta} = \int e^{itx} \ d\eta(x)$, and for a function $h: \mathbb{R} \to \mathbb{R}$ we have that the fourier ...
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9 views

Natural $\sigma$-algebra on the co-domain of a stochastic flow

Say that we have a stochastic local flow $X$. This is a map that sends $\omega$ into a collection of maps $\phi_{s,t}$ on $s,t\in \mathbb{R}$ which form a local flow. I don't understand how $X$ ...
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42 views

Finding the Cramer-Rao Lower Bound

Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound ...
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17 views

Formal definition of expectation question

I have expectation of a random variable on a probability space $(\Omega,\Sigma,P)$ defined as: If X is lebesgue integrable with respect to $P$ then $EX = \int_\Omega X \ dP$. What I don't understand ...
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33 views

Product of expectations is a martingale

Consider a probability space $(\Omega, \mathcal{F}, P)$ and random variables $X_0, X_1, \ldots , X_n$ adapted to the filtration $\{\mathcal{F}_t\}_{t\geq0}$. Assume furthermore that each $X_n$ is ...
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32 views

A question on conditional expectation leading to zero covariance and vice versa

In my probability class I was tackled with this seemingly weird question involving conditional expectation: Let X,Y be two random variables (it is not mentioned whether or not they are discrete or ...
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27 views

Transform Markov chain that doesn't have stationary transition probabilities to one that does?

This question concerns Exercise 7.3 in Walsh's Knowing the Odds. A Markov chain is defined as having stationary transition probabilities if for all $i, j, n$ we have $P(X_{n+1} = j \mid X_n=i) = ...
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30 views

An intuitive explaination..

I have a problem, to whose answer I want an intuitive explanation. Let $X_1,\cdots,X_5$ be independent continuous random variables having a common distribution function $F$ and density $f$. Let $I= ...