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
21 views

Probability ( kth order statistics <x , (k+1)th order stat >x)

consider $n$ iid draws according to some cdf $F$. What is the probability of the following event: the $k^{\text{th}}$ highest value is smaller than $x$ AND the $k+1$ highest value is larger than $x$. ...
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1answer
33 views

$X_i$ are Gaussian random variables with mean $0$ and variance $\sigma$

Let $X_1,...,X_k$, $k\geq 2$, be independent random variables, each having the same positive and differentiable density function $f$. Further suppose $\prod_{j=1}^k f(x_j)$ depends only on ...
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1answer
27 views

Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra

I am trying to prove that collection of all finite dimensional cylinder sets is an algebra but not $\sigma$-algebra. Cylinder sets are defined as: $\mathcal{B}_n$ is defined as the smallest $\sigma-$ ...
2
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0answers
23 views

weakly open subset in $M[0,1]$ (the space of finite measures on $[0,1]$)

I came across this question when reading Lynch and Sethuraman (1987): Large deviations for processes with independent increments. Let $M[0,1]$ be the space of finite nonnegative measures on $[0,1]$ ...
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1answer
38 views

Is distribution function always an event? [closed]

Is $\{\omega\in\Omega : X(\omega)\le x\}$ always an event? Does it always belong to the $\sigma$-field $F$ of the experiment?
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0answers
39 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
2
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0answers
10 views

Bound variance proxy of a subGaussian random variable by its variance

If I know $X$ is a sub-Gaussian random variable, and I know it has finite variance $\sigma^2$. Can I assert that $\sigma^2$ is a valid variance proxy for $X$? Definition (sub-Gaussian Random ...
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1answer
22 views

Expectation of time series

I have been given the following time series which has infinite history $X_t = 0.4X_{t-1} + 0.2X_{t-2} + \epsilon_{t} + 0.025$ where $\epsilon_t$ is white noise distributed $N(0,\sigma^2)$ First I ...
2
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4answers
81 views

How to calculate the probability of a point being inside a polygon [closed]

Given that a point is in a polygon, I am assuming that this point is more likely to be on (or near) the Centroid of the polygon than it is likely to be on (or near) the edges of the polygon. Is that a ...
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0answers
14 views

Intuition behind Stationarity in Delayed Renewal Processes

I was going through excess life and renewal processes in my notes when I came across a proposition in my notes that said that given a delayed renewal process X with independant interarrival times ...
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0answers
49 views

Conditional expectation random variable composed with a meas. function

I know that the following is true and fairly easily proven. Let $Y$ be a random variable and $\varphi$ a measurable function. Let $A$ be a $\Sigma_Y$ measurable set. If $ X (\omega) = \varphi(Y ...
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1answer
25 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuos stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
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1answer
40 views

Is this a measurable function

Let $\Omega_1 = \{ a, b, c, d \}$ and $Ω_2 = \{ 1, 2, 3, 4, 5 \}$ , and assume $F_i = \mathcal P ( \Omega_i ) ,\space i=1,2$. Consider a uniform probability assignment over $\Omega_1$ . For the map ...
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0answers
5 views

Countable additivity of a measure on the field of cylinders

In the 3rd edition of Bilingsley's Probability and Measure, there is this theorem (2.3), which says the following. Theorem: Every finitely additive probability measure on the field $\mathcal{C}_0 $ ...
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1answer
24 views

Doob decomposition of $ \sum\limits_{k \leq n}I_{A_{n}}$ [closed]

$\mathcal{F_{n}}$ is a filtration, where ${A_{n}} \in \mathcal{F_{n}}$, and $X_n = \sum\limits_{k \leq n}I_{A_{n}}$ What is the Doob decomposition for $X_n$?
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1answer
83 views

How to find expected angle between two randomly generated vectors?

Let us say two random points have been generated in a d-dimensional space by uniformly sampling from a unit cube centered at origin. How to calculate the expected angle between them?
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1answer
19 views

Weak Markov property implies strong Markov property for discrete time

From Klenke, p. 356: Theorem 17.14: If $I \subset [0,\infty)$ is countable and closed under addition, then every Markov process $(X_n)_{n\in I}$ with distributions $(\mathbf{P}_x)_{x\in E}$ has ...
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1answer
43 views

binomial distributions and their transforming (6.37-6.39)

I'm lost and frustrated. I don't know how the author (Karl Sigmund; The Calculus of Selfishness) transforms 6.37 in the book pages imaged below: $$ P_y = \sigma w^{N-1} + ...
3
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0answers
25 views

Estimation of the Hermite Polynomials using Plancherel-Rotach asymptotics

Suppose $H_n(x)$ is a Hermite Polynomial such that $$\int_{\mathbb{R}} H_n(x) H_m(x) e^{-x^2} dx = \delta_{m,n}.$$ I want to show for $ \phi_n(x) = H_n(x)e^{-\frac{X^2}{2}}$ $$ \left( ...
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0answers
12 views

the meaning of $f_+’(t) = \int_{(t,1]} {{s^{ - 1}}\mu (ds)}$

If $\mu$ is a probability measure on $[0,1]$, $$f_+’(t) = \int_{(t,1]} {{s^{ - 1}}\mu (ds)}\quad \mbox{for } 0<t<1.$$ Given $\mu(\{0\})=f(0+)$, how can we calculate $$\int_{(0,1]} {{s^{ - 1}}\mu ...
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0answers
21 views

Example of Markov process not strong Markov

This is again a question about this example (also see here). It seems we can write this process as $$X_t := \bigl(t - \text{Exp}(1) \mathbf{1}_{\{X_0 = 0\}}\bigr)^+ + \mathbf{1}_{\{X_0 \neq 0 ...
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1answer
31 views

Closure of the set of elementary predictable stochastic processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t)_{t\ge 0}$ be a real-valued stochastic ...
0
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1answer
26 views

Expected value of a function of a random variable [duplicate]

Let X be a random variable whose PDF is $f(x)$, and $g$ a function of random variable X. I want to prove that $$E[g(X)] = \int{g(x)f(x)dx} $$ I've perfectly understood it in discrete case and I ...
5
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1answer
42 views

Is the set of probability measures on a compact metric space (weak*-)closed?

Let $(S,\mathcal B)$ be a compact metric space with the Borel-$\sigma$-Algebra. Let $\mathcal M$ be the space of signed Borel measures and $\mathcal P \subset \mathcal M$ the set of probability ...
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0answers
29 views

Understanding the strong Markov property

I have problems to understand the strong Markov property (Klenke, p. 356): Let $I \subset [0,\infty)$ be closed under addition. A Markov process $(X_t)_{t\in I}$ with distributions $(\mathbf{P}_x, ...
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0answers
29 views

Proof of Levy's theorem without Ottaviani inequality

Suppose $X_1,⋯,X_n$ are independent r.v., Let $S_n=X_1+⋯+X_n$, I am looking to show that convergence on $S_n$ in probability implies almost sure convergence by showing that $P(\sup_{m\geq ...
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0answers
23 views

Upper and lower bounds on probability in binomial distribution.

Suppose i have a random variable $X \sim \mathrm{Bin}(n,p)$ and some $1 \leq l \leq n$ can i obtain good upper and lower bounds on the probability that $$\mathbb{P}(X \geq l)?$$ After some research I ...
0
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1answer
12 views

How to know the result of entropy function using uniform distribution set

In the entropy function here $H(s) = -\sum P(class=i|S)log_2{P(class=i|S)}$ I am trying to understand what is the domain of it's output for any input. I know that given a set where the frequency of ...
5
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3answers
185 views

What does actually probability mean?

I am a beginner in quantum information. Reading about it has made me question the definition of probability. If the probability of an outcome $m$ in an experiment is $p(m)$ then it means that if I ...
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1answer
18 views

Intro to probability chapter 4 ex 31

A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of pairs of people with the same birthday, ...
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0answers
71 views

Number of moves necessary to solve rubic's cube by pure chance

Suppose, random moves are made to solve rubic's cube. A move consists of a $90$-degree-rotation of some side. The starting position is also random. What is $E(X)$, where $X$ is the number of moves ...
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1answer
39 views

Exercise in measure theory/probability

This is an exercise in chapter 2 of Probability with Martingales by David Williams. Question: Let $\mathcal{A}$ be the set of all maps $\alpha : \mathbb{N} \rightarrow \mathbb{N}$ such that ...
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0answers
19 views

number of moves in the “memory”-game

Suppose, a player with a perfect memory starts with $2n$ cards in the game memory. A move consists of choosing two cards, the game ends if all pairs are found. Let $X$ be the number of moves the ...
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0answers
99 views

Sigma algebra generated by a homeomorphic random variable

Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every ...
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1answer
19 views

Given the distribution of $X$, whats the distribution of $cX$

Let's say $X \sim \chi_k^2(\lambda)$ with pdf $f_x(x)$ (i.e. noncentral chi-squared distribution). What can we say about the distribution of $Y = cX$ ? where $ c \in \mathbb{R}^+$ I know that $f_y(y) ...
2
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1answer
42 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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0answers
9 views

Moments bounds VS Chernoff bounds

I have to prove that, when bounding tail probabilities of a nonnegative random variable, the moments method is always better than the classical Chernoff method. In mathematical language, I have to ...
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1answer
26 views

No arbitrage iff there EMM $P^*$ theorem [closed]

The definition of an arbitrage I was given: "An arbitrage strategy is an admissible strategy with zero initial value and positive probability of a positive final value." I think that an initial ...
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0answers
42 views

Conditional density along a curve?

Let $X,Y$ be two random variables with $f_{X,Y}$ as their joint pdf and with $f_X(x)$ and $f_Y(y)$ as their marginal pdf's. The conditional pdf of, say $X$ with respect to $Y$ is $$f_{X\mid ...
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0answers
12 views

Interpretation of the Snell Envelope wrt European and American options.

In lectures we were taught that the Snell envelope U of a process Z, passes from European to American option prices. I thought that price processes are supposed to be martingales, but the Snell ...
3
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1answer
108 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda ...
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1answer
16 views

Formula for average time in Markov chain

I have a model like: A B C A 0.80 0.10 0.10 B 0.20 0.75 0.05 C 0.10 0.10 0.80 How do I get the average time from B to A? I understand that ...
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1answer
56 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
1
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1answer
11 views

conditional expectation conjugate exponents

If I know that $X\in L^p(\Omega,F,P)$ and $Y\in L^q(\Omega,G,P), \ G\subset F$, $\frac{1}{p}+\frac{1}{q}=1$, $F$ is $\sigma$ algebra on probability space $\Omega$, $G$ is sub $\sigma$ algebra. How ...
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171 views
+100

Conditional expectation, quadratic function, absolute value

We are given two random variables defined on $[0,1]$: $$X(\omega) = 2 \omega -1 + |2 \omega -1|$$ $$Y(\omega) = 1-|2 \omega^2 -1|$$ I am supposed to find $\mathbb{E}(X|Y)$ which by definition is a ...
0
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1answer
36 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
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16 views

Discrete measures converging weakly

Let $a_1, a_2, \ldots$ be any sequence of non-negative real numbers with $\sum_i a_i = 1$. Define the discrete measure $\mu$ by $\mu(\cdot) = \sum_{i\in\mathbb{N}} a_i \delta_i(\cdot)$, where ...
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1answer
15 views

Weak convergence equivalent to convergence of distribution functions

I'm trying to understand the following proof: I don't see where the Lipschitz property of the $f$ is needed. Why isn't bounded and continuous enough? Can somebody explain it to me? Thanks.
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0answers
46 views

Limit of an integral (Arrow theorem)

Im not sure about a limit of an integral. I would like to prove that there is a solution for this integral for d, and this solution is unique. The integral is: $$\beta = \int_d^{\infty}(x-d)f(x) dx$$ ...
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0answers
10 views

de Finetti's theorem in two dimensions?

We know that for an array of exchangeable Bernoulli r.v.s $X_i, i\in \mathbb{N}$, de Finetti's theorem can be rephrased to be that $$\exists f: \mathbb{R\times \mathbb{R}}\rightarrow \{0,1\}, \; ...