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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513 views

Definition of atomic $\sigma$-field.

Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
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1answer
81 views

How does Lindeberg CLT imply classicial CLT?

For a sequence of zero-mean $\sigma^2$-variance IID random variables $X_1, \dots,$ the Linderberg condition is applied this way: $\forall \epsilon >0$, as $n$ goes to $\infty$, $$\sum_{i=1}^n ...
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1answer
330 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
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42 views

Simple probability system

A system consists of four components ${1,2,3,4}$. There are two routes ${1,2,4}$ and ${1,3,4}$ to complete the system. If all four components have equal reliability of $0.9$, what is the reliability ...
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1answer
149 views

Is the product of two mixing random variables also mixing?

Question: Assume $X_t$ and $Y_t$ are random variables from the same probability space adapted to the filtration $\mathcal{F}_{-\infty}, ..., \mathcal{F}_t, ..., \mathcal{F}_{\infty}$. If $X_t$ and ...
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1answer
84 views

Relation between support of image-measure and closure of the image

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a measurable map. For a probability measure $\mathbb P$ denote by $\mu_\mathbb P$ the image measure of $\mathbb P$ ...
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2k views

Probability that numbers 1…6 show up at least once when rolling 8 dice

Probability that numbers 1...6 show up at least once when rolling 8 dice How can this be solved using the inclusion-exclusion principle.
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1answer
221 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
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146 views

Probability, Poisson Process, My solution is correct?

Jobs are submitted for a computer according to a Poisson process with a rate λ (jobs / hour). Determine the probability of at least two jobs are submitted within the 'a' first few minutes. I tried ...
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1answer
84 views

How is $P(|A) $ defined from $P(|\mathcal{B})$?

Let $ (\Omega, \mathcal A, \operatorname{P})$ be a probability space, with a real random variable $X$ and a sub-σ-algebra $ \mathcal B \subseteq \mathcal A$ In probability theory with measure ...
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1answer
122 views

Lévy-Khintchine formula for Cauchy distribution

The Lévy-Khintchine formula for the log of the characteristic function of an infinitely divisible random variable is $$ \Psi(s)=ias + \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} + ...
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73 views

Geometric distribution, tossing a die [duplicate]

Possible Duplicate: geometric distribution throwing a die Yesterday I posted a question which was answered but I disagree with the answer so I'd like to ask again so we can discuss it ...
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2answers
204 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
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2answers
256 views

Sequence of martingales

Let $(X_n)$ be a sequence of independent, identically distributed random variables with finite moment-generating function $M(t) = \mathbb{E}\left[\exp(tX_1\right)] < \infty$ for $t \in \mathbb{R}$. ...
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1answer
221 views

Poisson distribution and probability of random variables

Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier. Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
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1answer
155 views

Why is the following example of a Markov process not strong Markov

$X(t) := 0 \;\; (t \leq \tau),\;\; t - \tau\;\;(t \geq \tau)$ with $\tau$ exponentially distributed. Then X has the Markov property but not Strong Markov Property. But why ???? Can someone kindly ...
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326 views

Proof of $\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$

$X$ is a random variable defined in $\mathbb N$. How can I prove that for all $n\in \mathbb N$? $ \text E(X) =\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n ...
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1answer
3k views

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

Prove that if X and Y are Normal and independent random variables, X+Y and X−Y are independent. Note that X and Y also have the same mean and standard deviation. Note that this is a duplicate of ...
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1answer
87 views

independence between an event and a set of events

On a probability space, from Billingsley's Probability and Measure, a family of classes of events is said to be independent, if we arbitrarily choose an event from each class in the family, and ...
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1answer
250 views

A question about the expected number of games being played

refer to this question I wanna change the question, if the rule of the game is the same, what is the expected number played to end the game?
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1answer
244 views

Transforming an inhomogeneous Markov chain to a homogeneous one

I fail to understand Cinlar's transformation of an inhomogeneous Markov chain to a homogeneous one. It appears to me that $\hat{P}$ is not fully specified. Generally speaking, given a $\sigma$-algebra ...
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1answer
453 views

Expectation and Median (Jensen's inequality) of Spacial Functions

I hope this forum will be able to help me- if we have a 1-Lipschitz function $f:S^n \to \mathbb{R} $ , when $S^n$ is equipped with the geodesic distance d and with the uniform measure $ \mu $ . ...
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1answer
544 views

Does this condition imply the Lindeberg condition?

Given a double array of random variables $X_{nj}, j=1,\dots, k_n, n\in\mathbb{N}$ with $k_n \to \infty$ as $n \to \infty$, suppose for each $n$, $X_{nj}, j=1,\dots, k_n$ are independent, each ...
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3k views

Conditional Covariance of Functions of Random Variables

$\newcommand{\Cov}{\operatorname{Cov}} \newcommand{\E}{\mathbb{E}}$ I realize that $\Cov(X,Y) = \E[(X-\mu_X)(Y-\mu_Y)] = \E[\Cov(X,Y|A)] + \Cov(\E[X|A], \E[Y|A])$. But I am not sure how this is ...
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236 views

Question (2) on Uniform Integrability

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W) = 1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that ...
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497 views

Density of compound Poisson process

Is the probability density function (pdf) of the Compound Poisson $X(t)=\sum_{i=1}^{N(t)}Y$ known? Where $N(t)$ is a Poisson process and $Y$ is normally distributed with mean $\mu$ and variance ...
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112 views

Induction on Uniform Boundedness

This question gives interesting insights on whenever uniform boundedness can be "iterated". Let $\mu(\cdot)$ be a probability measure on the closed set $Z \subseteq \mathbb{R}^p$, so that $\int_Z ...
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1answer
218 views

From bounded to unbounded set in Lebesgue integration

Let $\mu$ be a probability measure on $X$. Consider a family of functions $\phi_k: X \rightarrow \mathbb{R}_{\geq 0}$ such that $\sup_k \phi_k(\cdot)$ is integrable over $X$. Let $\{X_n\}$ be an ...
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2answers
95 views

Help in Independence of events

Question: For events $A_1, A_2, \ldots, A_n$ consider the $2^n$ equations $P(B_1\cap\ldots\cap B_n)=P(B_1)\ldots P(B_n)$ with $B_i=A_i$ or $B_i=A_i^c$ for each $i$. Show that $A_1,\ldots,A_n$ are ...
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1answer
638 views

Definition of random variable, Borel $\sigma$-algebra

I´m a beginner in more advanced probability and measure theory and there is this definition that I simply can´t understand. It says, a random variable is a function $X\colon\Sigma\to \mathbb R$ with ...
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1answer
404 views

Conditions for the disintegration theorem to hold

The disintegration theorem says that under certain conditions, a probability measure $\mu$ on a measurable space $$ the existence of Let $Y$ and $X$ be two Radon spaces (i.e. separable metric ...
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1answer
617 views

Expectation product of pairwise uncorrelated variables

Suppose I have three uncorrelated random variables $X, Y$ and $Z$ (discrete or continuous) such that $$\newcommand{\Cov}{\mathrm{Cov}}\Cov(X,Y)=0;\quad \Cov(Y,Z)=0;\quad \Cov(X,Z)=0 \tag{$\ast$}$$ I ...
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1answer
875 views

Law of Large Numbers and Cauchy Distribution

Let $\{X_n\}_{n=1}^\infty$ be a sequence of iid random variables under the Cauchy distribution with location parameter $0$ and scaling parameter $1$ (so that the density function is $f(x) = ...
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1answer
164 views

Is $\sinh(W_t+t)$ a submartingale, and the expected value

Suppose I have $X_t=\sinh (W_t+t)$. I am not sure how to show if this is a submartingale, and how to calculate its expectation. I don't want to integrate this against the normal distribution to find ...
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1answer
436 views

“Physical” meaning of higher moments (their values and their existence)

Suppose I have a probability distribution $A$ with continuous support over $\mathbb{R}$. Suppose $A$ has a sequence of finite (central) moments $\mu_1, \mu_2,\ldots,\mu_n$. I understand that $\mu_1$ ...
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2answers
151 views

Central limit theorems, Almost sure invariance principles and Brownian motion

In a paper I was reading on dynamics, I came across a proof of a central limit theorem in a certain situation using brownian motion and an almost sure invariance principle. I am not very experienced ...
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2answers
211 views

Calculating probabilities on a spherical map

A black and white colored sphere is given. We are looking at a random starting point on the sphere below us, which has a certain color. A random rotation can change the color of the spot below us. ...
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1answer
224 views

Elementary and measure definitions of conditional expectation and probability

In elementary probability, $E(Y \mid X =x)$ is defined as expectation of $Y$ w.r.t. the p.m. $P(A \mid X =x): = \frac{P(A \cap \{X=x \})}{P( X=x)}$ when $P( X=x) \neq 0$. ...
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1answer
248 views

Question for understanding definition of point process

I am trying to understand the definition of point process when reading its Wikipedia article: Let $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra ...
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1answer
455 views

Definition of conditional expectation of a random variable given another one

Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $(U, \mathcal{\Sigma})$ is a measurable space. There seem to be two ways of defining the conditional expectation of a r.v. $X: \Omega ...
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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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32 views

Transition function for absorbed Brownian motion

I need an help with the following exercise. I've already seen this question Prove that Brownian Motion absorbed at the origin is Markov but I don't understand the answer. Also I would like to prove ...
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1answer
38 views

product of two multivariate normal densities for the same vector, if one is only specified for a subset

A random vector x with n elements has a multivariate-normal density f(x). Another distribution is known for m linear combinations of elements of x. The linear combinations are given in the form ...
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94 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
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17 views

Cumulative beta density calculation. [duplicate]

The beta distribution of y, w.r.t $\alpha,\beta, min - a \text{ and } max - c $ is. $$f(y; \alpha, \beta, a, c) = \frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)}$$ ...
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22 views

Martingale with bounded increments converges or diverges to $\pm \infty$ [duplicate]

Let $(M_n)$ be a martingale with $|M_n - M_{n-1}| \leq c$ for some fixed $c < \infty$. Check that the two disjoint events $$C:=\{M_n \text{ converges to a finite limit}\}, \; F:=\{\limsup M_n ...
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20 views

Probability calculation when the R.V. are the ratio of variables i.i.d. and exponentially distributed

Suppose to have two R.V. i.i.d., i.e. $Z$ and $T$, given by \begin{equation} Z = \frac{X}{1+Y}, \end{equation} and \begin{equation} T = \frac{U}{1+V} \end{equation} I have to calculate the ...
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1answer
38 views

Exercise about conditional expectation

I have to show that for sigma-algebras $\mathcal{F, G}$ with $\mathcal{F}\subseteq \mathcal{G}$ and $X, Y$ real random variables with $\Bbb E[X^2] < \infty$ the following holds: ...
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18 views

Show if $p^n(i,j)\rightarrow \pi(j)$ as $n\rightarrow\infty$ then $\pi(j)$ is a stationary measure..

Suppose $p(i,j)$ is a transition kernel on $S$ for a countable state markov chain $X_n$ with $$p^n(i,j)\rightarrow \pi(j)$$ as $n\rightarrow\infty$ for all $i,j\in S$. want to verify that $\pi$ is a ...
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1answer
34 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...