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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3
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0answers
34 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
0
votes
2answers
33 views

Find dependent event when two dice are thrown simultaneously.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of ...
0
votes
0answers
25 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
1
vote
0answers
50 views
+50

Show $P\left[A<Z \mid \mathcal{G} \right]=e^{-A}$ for $Z$ standard exponential and $A$ nonnegative $\mathcal G$-measurable

I have a question about exponential distribution and conditional probability. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{G}$ be a sub $\sigma$ algebra of $\mathcal{F}$. Let ...
0
votes
1answer
28 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
1
vote
2answers
31 views

Probability Question - Moment Generating Function

$$ f(x) = \begin{cases} xe^{-x}, & \text{x ≥ 0} \\ 0, & \text{elsewhere} \end{cases}$$ Q: Find the Moment Generating Function of X. Hi, I was trying to solve this question by putting the ...
2
votes
3answers
58 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
2
votes
0answers
31 views

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
2
votes
3answers
54 views

What is a sample of a random variable?

I've tried to learn probability, one way or another, many times in the last 50 years, and finally settled on the Kolmogorov approach, where a random variable isn't described as something like "a roll ...
0
votes
0answers
8 views

Proposition in Daniel Kuhn et al paper “Primal and dual linear decision rules in stochastic and robust optimization”

Paper link:http://www.optimization-online.org/DB_FILE/2009/02/2218.pdf In this paper in page number 8 the autors make the following proposition: Let $\mathbb{P}$ be a probability measure on ...
2
votes
1answer
13 views

Every collection of measures on a compact space is uniformly tight

I am looking for a proper statement of the sentence in the title and its proof. First, let me give some context. I have a covariance stationary time series, $X$. The autocovariance function of $X$ is ...
-3
votes
0answers
19 views

Function of two continuous random variables. find CDF [on hold]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
2
votes
0answers
18 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
-1
votes
1answer
39 views

Let X be a continuous random variable with pdf… [on hold]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...
0
votes
1answer
13 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, ...
0
votes
0answers
13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
0
votes
0answers
10 views

Connect the MGF of the Survivor, Cumulative and Mass disttributions

Assume that $X$ has a known distribution $P_X$, with a generating function $\hat P_X$. What relationship links $\hat P_X$ with the MGF of X's CDF ($\hat C_X$) and SDF ($\hat S_X$). Would that ...
0
votes
2answers
49 views

Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$

Let $f_X(k) = f_Y(k)= p(1-p)^k~$ for all $k = 0,1,2,\ldots$ for some $0 < p < 1$. Show that for any $n \ge 0$ $$P(X=k \mid X+Y=n) = \frac1{n+1}$$ for any $0 \le k \le n$. What is confusing ...
0
votes
0answers
9 views

Deriving the asymptotic properties of two-stage estimators

Suppose $Y_n = g(X_1, \cdots, X_n)$ is a statistic and that $\sqrt{n}(Y_n -\theta) \stackrel{d}{\to} N(0,V)$, where $\theta$ is a constant. Define $Z_{i,n} = f(X_i,Y_n)$, where $f$ is a continuously ...
0
votes
0answers
29 views

Dependence/Independence Problem in probability [on hold]

Suppose a student is taught by $N$ teachers. The pdf of the marks that the student gets from $i$-th teacher is $p_{m_i}(x)$ and we assume that all $p_{m_i}(x)$'s are i.i.d i.e. ...
-6
votes
0answers
77 views

Bernoulli Process [on hold]

Customers depart from a bookstore according to a Bernoulli process with parameter p = 0.15 (per minute). Each customer buys a book with probability 2/3, independent of everything else. Find the ...
0
votes
1answer
37 views

If I randomly select 6 books, what is the probability I…

I have 30 books. 5 are labeled classics, 10 are labeled mysteries, 7 are labeled science, and the rest are sports. If I randomly select 6 books, what is the probability I a) select at least 2 ...
1
vote
0answers
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 ...
0
votes
2answers
30 views

Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. $$ I tried using integration by parts to obtain \begin{align} ...
1
vote
1answer
22 views

An equilateral triangle has one vertex at the origin of $Oxy$ plane, one vertex at $(1;0)$, and one in the 1st quadrant. Find $cov(X;Y)$

An equilateral triangle has one vertex at the origin of $Oxy$ plane, one vertex at $(1;0)$, and one in the 1st quadrant. Suppose you choose one of these three vertices uniformly at random (i.e. each ...
2
votes
0answers
15 views

Estimate for Expectation of Reciprocal Bessel Process

Let $W=(W_{t})_{t\geq 0}$ be a standard $3$-dimensional Brownian motion, and let $a\neq 0\in\mathbb{R}^{3}$. Consider the $3$-dimensional inverse Bessel process defined by ...
0
votes
1answer
10 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
0
votes
0answers
39 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ ...
-1
votes
1answer
34 views

ergodicity in $\mathbb{Z}^d$

Fix $d \geq 1$ and let $E(\mathbb{Z}^d)$ denote the set of all edges of the graph $\mathbb{Z}^d$. Let us consider a measure preserving system $(\mathbb{R}^{E(\mathbb{Z}^d)}, ...
1
vote
1answer
34 views

Bernstein Inequality (wiki correction?!)

I am having trouble with one of the statements made on this wikipedia page, in particular the second Bernstein Inequality on: https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) ...
0
votes
0answers
35 views

How to find approximate probability of obtaining population variance between $10$ and $15$? [on hold]

A sample of $15$ observations is taken from a normal population. It has been calculated that the sample mean is 30 and the sample variance is $12.1$. Find the approximate probability of obtaining ...
0
votes
2answers
20 views

measure preserving system

Let $T$ be a measure-preserving transformation on a probability space $(\Omega, \mathcal{F}, P)$ and let $A \in \mathcal{F}$ such that $P(A) > 0$. (i) Show that there exists $n \geq 1 $such that ...
1
vote
1answer
46 views

Finding pdf of the distance between two points (My strategy right or wrong?)

I have $N$ points randomly distributed in between points $A$ and $B$ in an area. I want to find the pdf of the distance between point $A$ and $B$. Prior Knowledge: 1- $f_{d_{A,i}} \forall i \in ...
1
vote
0answers
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 ...
4
votes
1answer
30 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
0
votes
1answer
35 views

expected value of fisher distribution

I know that the pdf of an F-distribution is $f_{k,m}(t) = \Gamma(t)=\frac{\Gamma((k+m)/2)}{\Gamma (k/2)\Gamma(m/2)}k^{k/2}m^{m/2}t^{k/2 - 1}(m+kt)^{-(k+m)/2}$ Also, $E(F)=\int xf_{k,m}dx$. How do ...
2
votes
1answer
42 views

Does weak convergence ($X_n\Rightarrow X$) imply weak convergence of the difference to zero ($X_n-X\Rightarrow 0$)?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am wondering if this implies that $X_n-X\Rightarrow 0$ as $n\rightarrow ...
2
votes
0answers
14 views

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
0
votes
1answer
26 views

how to calculate conditional independence

This Bayesian net (click) is given with the binary variables B, F, G and D and the following probabilities $p(B=1) = 0.9$ $p(F=1) = 0.9$ $p(G=1\mid B=1,F=1) =0.8$ $p(G=1\mid B=1,F=0) = 0.2$ ...
2
votes
3answers
26 views

Proving continuity of the following function

Let $X,Y$ be compact sets in $\mathbb{R}^n$ (with the usual topology) and let $f:X\times Y \rightarrow \mathbb{R }$ be a continues function function moreover let $P(Y)$ be the space of all ...
1
vote
0answers
14 views

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 ...
1
vote
1answer
31 views

Conditional probability involving coin flips

A coin has an unknown head probability $p$. Flip $n$ times, and observe $X=k$ heads. Assuming an uniform prior for $p$, then the posterior distribution of $p$ is $B(\alpha = k + 1, \beta = n - k + ...
1
vote
1answer
26 views

Probability proof and making sure I cannot make further simplifications to my answer

The question asks me to compute the probability (sums are ok) of the probability of having at least one of r cells empty with n>r balls thrown at the cells with equal likelihood of landing in any of ...
1
vote
2answers
17 views

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Suppose X and Y are integrable random variables on the measure space $(\Omega,\mathcal F, P)$. Im trying to show that $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$ but I got ...
1
vote
0answers
17 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 ...
0
votes
0answers
14 views

Calculating the Distribution Function of Chord Length

The question: Choose two (different) points on the circle $S^1 \subset \mathbb{R}^2$ at random (with uniform distribution), and connect them with a straight line. Define a suitable probability space ...
2
votes
0answers
21 views

I found $E(Σ_{j=0}^{k-1}η_j-Σ_{j=0}^{k-1}E(η_j|G_j))^2=Σ_{j=0}^{k-1}(E(η_j)^2-E(E(η_j|G_j)^2)$ in a book with faulty assumptions on the objects

In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following: If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and ...
0
votes
0answers
26 views

Upper bound for $\frac{\|x\|_1}{\|x\|_2}$ if each entry of $x\in R^d$ is i.i.d. sampled from Gaussian distribution $N(0,1)$

In the question, $\|x\|_1=\sum_{i=1}^d|x_i|$ with $|\cdot|$ being the absolute value, and $\|x\|_2=\sqrt{\sum_{i=1}^d x_i^2}$. In general, $\frac{\|x\|_1}{\|x\|_2}\leq \sqrt{d}$ always holds for ...
1
vote
1answer
22 views

How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
0
votes
1answer
17 views

Moment generating function properties: $3φ_X (t)$ and $φ_X (t) × φ_X (6t)$

Suppose that $φ_X (t)$ is the moment generating function of some random variable $X$. Are the following functions moment generating functions of some (other) random variables? i. $3φ_X (t)$ I think ...