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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Proof of the Box-Muller method

This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly ...
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1answer
35 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
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1answer
37 views

What are the possible values of Z=X+Y

If I have two independent probability mass function, where $P_{x}(0)=\frac{1}{2}$ , $P_{x}(2)=\frac{1}{2}$ and $P_{y}(1)=\frac{1}{6}$ , $P_{y}(2)=\frac{1}{3}$ , $P_{y}(3)=\frac{1}{2}$ I am asked ...
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“Expected value” of Thirteen card game !! [duplicate]

Thirteen cards numbered $1$ to $13$ are shuffled and dealt one at a time. "Match" occurs on deal $k$ if $k$th card revealed is card number $k$ Let N be the total number of matches that occur in the ...
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1answer
33 views

Convergence in distribution conditions

I'm reading the weak convergence section (Ch.$18$) in "Probability Essentials" by Protter and Jacod. Theorem $18.7$ states that "$Xn \overset{D}{\to} X \iff \underset{n\to\infty}{\lim} E[g(X_n)] = ...
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83 views

Almost surely either $X_n=0$ for some $n$ or $\lim_{n\to\infty}X_n=\infty$

How I came to this: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of non negative random variables, $\mathcal{F}_n=\sigma(X_l,l\leq n)$ the sequence of corresponding sigma-algebras and define ...
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35 views

Calculate probability of nodes in a graph

I have the following graph: A and F post the same joke on Facebook. Now there is a probability of 0.6 that node b will post the joke too. and so on... So the weights on the edges say how likely it ...
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18 views

Example where the disintegration theorem does NOT hold

There are various conditions which guarantee that the disintegration theorem will hold, i.e. a joint probability distribution over a product space may be decomposed as a marginal probability and a ...
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2answers
143 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
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1answer
44 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
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1answer
126 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
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34 views

Calculating $\mathbf{P}[X < Y]$ for $X, Y$ exponentially distributed?

This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course« Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain ...
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1answer
20 views

An equality from the well-known analysis of variance formula

Suppose that we have a parametric model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ dominated by some measure $\mu$. That is, each $\theta$ is associated with a density $l(y;\theta)$. Let $S(Y)$ ...
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31 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}-\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...
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64 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
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1answer
27 views

Unbounded stopping time

Suppose we have a sequence of i.i.d. random variables $(X_n)_{n \in \mathbb{N}}$ with $\mathbf{P}(X_n = -1) = \frac{1}{2}, \mathbf{P}(X_n = 0) = \frac{1}{3}, \mathbf{P}(X_n = 1) = \frac{1}{6}$. Denote ...
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1answer
25 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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72 views

Exercise in Probability/Measure Theory

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let also $A_{n,j}\in\mathcal{F},n\in\mathbb{N}_0,j\in\{1,2,3,...,2^n\}$, be such that for all ...
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36 views

Create a Martingale out of a Markov Chain.

Consider a homogeneous finite state Markov chain $\{X_n\}$ with transition matrix $P$ and state set $S$ consisting of real numbers. How to choose the elements of $S$ so that $\{X_n\}$ be a ...
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20 views

About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
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0answers
20 views

Convergence in probability of the Fisher information

Given a family $\{\mathbb{P}_\theta\}_{\theta\in\Theta}$ on $\mathcal{B}(\mathbb{R})$, where $\Theta\subset\mathbb{R}$ and each member of this family is absolutely continuous w.r.t. $\lambda^1$, and ...
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1answer
28 views

Does law of total probability apply here?

Let $X$ and $Y$ by positive independent random variables. Let $f(x,y)=\frac{ax}{y^2+ay}-\frac{ab}{y}$ where $a>0$ and $b>0$ are constants. I am wondering if the following is true: ...
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1answer
24 views

Condition in a theorem in Probability theory.

I passed by a simple theorem in Probability theory , yet it really bugs me that I think that 1 condition in the hypothesis is not necessary. After checking the proof for many times, I still can't ...
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1answer
165 views

Expected value of a complicated function of more than one random variable.

Assume we have random variables with Probability Density Functions (pdf) as follows $$\omega_i \sim f_{1},\,\,\,\,\ i \in[1:n]$$ $$ \gamma= \{\gamma_1,\cdots,\gamma_n\} \sim f_2: \text{joint pdf of ...
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17 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
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1answer
22 views

PDF of distance from the center of a random point in the unit disk

if find in a certain website ( also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: P(d)=2d, but no one is giving the way to derive ...
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1answer
64 views

Exercise on Martingales

I have been struggling with the following exercise and I was wondering whether my solution is correct or not. I am pretty sure about the second part of the question (the martingale part) but not so ...
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28 views

Question about Lebesgue Dominated Convergence Theorem involving a Markov Time / Stopping Time

I am trying to understand the proof of the following lemma: Let $W$ be an arbitrary random variable satisfying $\mathbb{E}[|W|] < \infty$, and let $T$ be a Markov time (or stopping time) for which ...
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1answer
27 views

Dependence of RVs exponentially distributed

Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in ...
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1answer
26 views

Does $P(X\leq a) = P(X^2\leq a^2)$ if $X$ is a positive random variable and $a>0$?

The answer looks positive to me, since $$P(\omega:X(\omega) \leq a) = P(\omega:X(\omega)^2\leq a^2)$$ Am I right?
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1answer
33 views

Characteristic function, how to integrate?

Find the characteristic function $\phi_X(t)$ of an absolute continious r.v. $X$ with density: $$f_X(x) = \frac{a}{2}e^{-a|x|} \qquad (a>0; x\in \mathbb{R})$$ Notation I have some ...
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24 views

given distribution function find the density

I am confused about one trivial thing. We have independent variables $X,Y,\beta$ where $X$ has a distribution function $F(x)$, $Y$ has $G(x)$ and $\beta$ is $Bin(1,p)$ distributed. Also $X,Y$ have ...
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Proof regarding cumulative distribution functions…

Here is the theorem: Theorem The following statements are equivalent: a. The random variables $X,Y$ are identically distributed b. $F_X(x)=F_Y(x)$ $\forall x$ In our book, they ...
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Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
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1answer
48 views

Limit of measures is a measure

I know the following theorem (see exercise 1.3.3 from Achim Klenke: »Probability Theory — A Comprehensive Course«): Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of finite measures on the ...
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32 views

What is the interval for an exponential random variable?

Suppose that I have generated a random number $x$ using an exponential distribution with rate parameter $\lambda$. How can I find an interval $[a,b]$ such that $x$ is in this interval with probability ...
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5answers
128 views

Conditions for uniqueness of the median

A median of a random variable is defined as any $m \in \mathbb{R}$ such that $P(X \le m) \ge 1/2$ and $P(X \ge m) \ge 1/2$. Alternatively, in terms of the CDF $F$ of $X$ defined by $F(x) := P(X ...
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1answer
22 views

Conditional probability -conditioning on a random variable

Let $ (\Omega, \mathcal{F}, \mathbb{P}) $ be a probability space, $A \in \mathcal{F}$ and $X$ a random variable. What does it mean $$ \mathbb{P} (A | X) $$ when $X$ is not discrete? Thank you!
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Does this sequence converge almost surely or not?

I have a sequence of independent random variables $X_1, X_2,...$ such that $P(X_n = 1) = \frac{1}{n}$ and $P(X_n = 0) = 1 - \frac{1}{n}$. Using the second Borel-Cantelli Lemma, we have $\sum P(X_n ...
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1answer
19 views

Clarification - DeGroot Proof on Transitivity Property of Subjective Probability

In developing axiomatic foundation for subjective probability DeGroot (Optimal Statistical Decision, 2004, p71) gives two axioms/assumptions: SP1: For any two events A and B, exactly one of the ...
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3answers
72 views

Probability that a size $d$ sample will contain all $k$ colours present

I tried looking for this question, but couldn't find it exactly... apologies if this is a repeat! Imagine an urn with $m$ balls. Each ball has a different colour and there are $k$ colours (obviously, ...
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1answer
20 views

Expectation of maximum of Binomial RVs

Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find $$E(X_{(n)})$$ that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that ...
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27 views

Finding probabilities - Elementary probability.

Suppose $\mathbb P(A) = 0.3$, $\mathbb P(B) = 0.5$, and $\mathbb P(B |A) = 0.6$. a. Find $\mathbb P(A \text{ and } B)$. b. Find $\mathbb P(A \text{ or } B)$. c. Find $\mathbb P(A|B)$. ANSWER: a. ...
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1answer
38 views

Conditional independence of sigma-algebras

If ${\mathcal{H}_1}$ and ${\mathcal{H}_2}$ are conditionally independent given $\mathcal{G} \subseteq {\mathcal{H}_2}$, are they conditionally independent given $\mathcal{F}$ such that $\mathcal{G} ...
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1answer
28 views

What does (0+) mean?

I'm currently learning from a script (which is written in German and not publicly available, sorry) for introduction to stochastics, where the topic is the Laplace transformed function for random ...
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1answer
31 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...
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37 views

A maximal inequality on distance to median, so called Lévy's inequality?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.3, ex6) Suppose $X_1,\dotsc,X_n$ are i.i.d. random variables, and $S_j:=X_1+\dotsb+X_j,S_j^0:=S_j-m_0(S_j)$, where $m_0(S_j)$ is a ...
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1answer
9 views

Expectation of distance from centre of a circular scattering

A random point $(X,Y)$ has a normal distribution on a plane with circular scattering with $E[X]=E[Y]=0$ and var$[X]$=var$[Y]$=$\sigma^2$. The distance of the point $(X,Y)$ from the centre of ...
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27 views

Probability Of 2 consecutive dice throw for 4 players

Consider a fair dice. 4 players are throwing the dice one after another. If some one gets 6 he will get extra chance to throw it again. So he will throw it twice and then the next player will get the ...
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45 views

Chebyshev's Inequality and the length of a random vector

Suppose that we take $n$ iid random variables $X_1,\dotsc, X_n\sim \operatorname{Unif}[-1, 1] $ and define $Y_n=\lVert (X_1,\dotsc, X_n) ^\intercal \rVert $. By what means can we employ ...