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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2
votes
1answer
65 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...
3
votes
0answers
15 views

Is it possible to build a new probability space so that the product of two independent Gaussian r.v. still be Gaussian in the new space?

I know that if $X$ and $Y$ are two independent normal random variables defined on the same probability space ($\Omega$, $\cal{F}$,$\cal{P}$), the product may not be normal, but is it possible to ...
3
votes
0answers
12 views

What is the limit distribution of $\frac{S_{N_n}}{\sigma \sqrt{a_n}}$ as $n\rightarrow \infty$.

Let $X_1, X_2, X_3,...$ be iid with $\mathbb E[X_i]=0$ and $Var[X_i]=\sigma^2>0$, and let $S_n = \Sigma_{i=1}^{n} X_i$. Let $N_n$ be a sequence of integer valued random variables independent of ...
0
votes
1answer
122 views

Is it true that $\|XY\|_r \leq \|X^r\|_p^{1/r} \|Y^r\|_q^{1/r}$ for $r>1$?

Given random variables $X$ and $Y$, Holder's inequality states that: \begin{equation} ||XY||_1 \leq ||X||_p ||Y||_q , \end{equation} for $\frac{1}{p} + \frac{1}{q} = 1$, and $p,q \in [1, \infty]$. ...
2
votes
2answers
23 views

Random variable independent of $\sigma$-algebra and conditional expectation

What does it mean to say that a random variable is independent of a sigma-algebra, and why then does this imply that $E(RV| \sigma) = RV$?. I have no clue what this independence stuff is about ...
0
votes
0answers
15 views

Elements contain in a sigma algebra generated by a set of random variables

Hello and thanks for the time spend to read this :) Consider $(\Omega,\mathcal{F},P)$ Consider $A=\{x_1,...,x_p\}$ a set of random variables and $\Theta=\sigma(A)$ be the sigma algebra generated by ...
-1
votes
1answer
43 views

Finding the conditional probability

enter image description here Let $(X,Y)$ be a two-dimensional stochastic vector with density $$ f_{X,Y}(x,y) = \begin{cases} \dfrac{e^{-y}} y & \text{if } 0<x<y, \\[4pt] \,\,\,\, 0 & ...
2
votes
2answers
98 views
+200

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
1
vote
0answers
38 views

How to model my not knowledge of distributions. Example form Optional Stopping.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
13
votes
4answers
2k views

Formally, why does a logical contradiction have probability zero?

In terms of formal probability theory, why does an event representing a logical contradiction (such as $A \wedge \neg A$) always have probability zero? I understand intuitively why this is the case, ...
0
votes
0answers
6 views

How to estimate the duration of the path?

Let $G=<V,E>$ $p$ - sequence of vertices and edges For each edge $(u,v)\in E$ there is information about the transition time from $u$ to $v$ represented as a set of values $T=\{t_0, t_1, ... ...
1
vote
1answer
13 views

Calculate the number of paths found on the basis of probabilities

Let G=<V,E> - weighted directed graph $w(e_{ij})$ - transition probability from node $v_i$ to node $v_j$ ($w \in[0;1]$) So my first question is: how ...
3
votes
0answers
35 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
1
vote
0answers
43 views

Zero-one law in binomial random graph model $G(n,p)$

Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...
0
votes
0answers
9 views

Does marginalizing on a Bayesian network preserve its original independence assumptions?

I know that marginalizing over a Bayesian network causes changes to the graph (e.g. marginalizing node $c$ in the V-structure given by $a \rightarrow c \leftarrow b$ results in $a$ and $b$ being ...
1
vote
0answers
16 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
1
vote
0answers
36 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ...
0
votes
0answers
24 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
-3
votes
0answers
24 views

Connectivety of the Erdős–Rényi random graph [on hold]

Let G be a graph in G(n, p) (Erdős–Rényi model) I want to prove that that P( G(n, p) where p ≥ ( lnn/10n) and number of tree components on 11 vertices = 0 ) converges to 1 and lnn/n is a ...
2
votes
1answer
55 views

Necessary conditions for unique convergence of a sequence of random variables.

Suppose that I have a sequence of random variables $\{X_n\}_{n\geq 1}$, where $X_i \in R$ for all $i=1,...,n$, for some space $R$. Furthermore, suppose I have a function $S$ with arguments in $R$ such ...
1
vote
0answers
24 views

Slutsky, Continuous mapping for uniform convergence

I have a question- suppose I have a function f(x,$\hat \theta$), $\hat \theta$ is a consistent estimate for $\theta$ and therefore it holds $\hat \theta \rightarrow \theta$ in probability. Suppose f ...
0
votes
0answers
14 views

Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
1
vote
1answer
20 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with ...
-1
votes
0answers
15 views

Finding joint pdf from marginal pdf's

I have $N$ samples $(X_1,\cdots X_N)$ of exponential random variables with parameter 1. The samples are ordered such that $X_N \geq X_{N-1} \geq \cdots X_1$. I know the individual pdf's of $X_N$ and ...
2
votes
0answers
23 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
0
votes
0answers
17 views

System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
-1
votes
2answers
53 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
2
votes
1answer
39 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
0
votes
1answer
25 views

If A is a Borel set in the real line and x is any real number, show that translate of A, defined by A + x = {y + x : y ∈ A} is also a Borel

If $A$ is a Borel set in the real line and $x$ is any real number, show that translate of $A$, defined by $A + x = \{y + x : y \in A\}$ is also a Borel set. Show that $−A = \{−y : y \in A\}$ is also a ...
0
votes
0answers
48 views

Random walk visiting $k$ distinct points

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
1
vote
0answers
45 views

Cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$?

I am trying to argue that the cumulative distribution function of two independent and uniform on $[0,1]$ random variables is a surjective map for $t\in [0,1]^2$. Below the argument I have developed. ...
5
votes
0answers
33 views

Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk ...
1
vote
1answer
26 views

What is the domain of a function of random variables?

Consider a random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, P)$ $X:\Omega \rightarrow \mathcal{X}\subset \mathbb{R}$. Suppose $X$ has range (or image) $\mathcal{I}\subset ...
3
votes
0answers
34 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
3
votes
1answer
54 views

Tic Tac Toe: What is the probability that a random player draws against an infallible player?

I have simulated a tournament between an infallible Tic Tac Toe player and one that chooses its moves randomly. Even after 5 million games, the infallible player wins every single game. I know that ...
0
votes
2answers
24 views

Dice role: What is the probability to observe 2 times 1 and 2 times 5 with the outcome of a fifth die role being unknown?

I tried to solve the following exercise: Given a dice with $P(X=2) = P(X=4) = P(X=5) = \frac{2}{15}$ and $P(X=1) = P(X=6) = P(X=3) = \frac{2}{10}$. What is the probability to observe 2 times 1 and 2 ...
0
votes
0answers
15 views

Probability of non-linear transformation

I'm reading about the accept-reject algorithm to generate non-uniform random numbers from the uniform. Let $X$ have a density on $\mathbb{R}^d$, and let $U$ be independent uniform on $[0,1]$. Then ...
0
votes
0answers
24 views

Difference of dependent central Chi-Square random variables with 2 degrees of freedom

Suppose we have $X$ and $Y$, both are dependent and complex Gaussian random variables with zero means and the same variance $\sigma^2$. The real and imaginary parts of $X$ and $Y$ are independent, ...
-1
votes
0answers
49 views

$X_n$ doesn't converge to a limit in $[-\infty, \infty] \to$ Is this supposed to be a stronger version of $\lim X_n$ doesn't exist?

From Williams' Probability with Martingales: What's the difference between saying that '$X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$' and '$\lim X_n$ does not exist' ? ...
1
vote
0answers
17 views

Intuition behind Mutual independence of sub-$\sigma$-algebras definition.

I was reading about Independence of sub-$\sigma$-algebras when I found the next definition: Let $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ $n$ sub-$\sigma$-algebras of $\mathcal{A},$ let $H$ be a ...
0
votes
0answers
25 views

Expected value of n-step Markov chain

Let $(X_k)_{k=1}^n $have distribution $$P(X_i = a_i | i \in {1, ... n}):= c \cdot\exp\left(\sum_{i=1}^{n-1} a_ia_{i+1}\right)$$ for $a_i \in \{-1,1\}$ and for $c$ a normalizing constant. I'm having ...
1
vote
1answer
50 views

Let X and Y be independent random variables such that $E|X+Y|<\infty$. Is it true that $E|X|<\infty$? Give a proof or a counterexample.

Let X and Y be independent random variables such that $E|X+Y|<\infty$. Is it true that $E|X|<\infty$? Give a proof or a counterexample. Thoughts: My intuition was to apply Fubini-Tonelli here ...
4
votes
0answers
21 views

Exploiting the Markov property

I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs. Let ...
2
votes
1answer
535 views

Formula similar to $EX=\sum\limits_{i=1}^{\infty}P\left(X\geq i\right)$ to compute $E(X^n)$?

Is there a formula like $$ EX=\sum_{i=1}^{\infty}P\left(X\geq i\right) $$ (which can be found on Wikipedia and holds for positive $X$) for $EX^{n}$ ? And I don't mean this one, $$ ...
0
votes
1answer
23 views

Convergence a.s. and convergence in $L^1$ don't imply each other [on hold]

I'm trying to get two examples that convergence a.s. and convergence in $L^1$ don't imply each other. Now, I only know the examples that convergence a.s can't implied by convergence in probability, ...
2
votes
0answers
17 views

Difference modes of convergence of a sequence of independent Bernoulli random variables

Suppose $(r_n)_{n \geq 1}$ is a sequence in $(0,1]$, $(X_n)_{n \geq 1}$ is a sequence of independent Bernoulli random variables such that: $P(X_n=0) = 1 - r_n, P(X_n = \frac{1}{r_n}) = r_n$. Show ...
0
votes
0answers
32 views

Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
3
votes
1answer
36 views

Limit Brownian Bridge Integral

As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for $0 \leq t <1$. In order to show that for any $g ...
1
vote
1answer
51 views

Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
1
vote
0answers
45 views

Version of the CLT

It is well known, that for a sequence of i.i.d. rv. $X_i$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^{2}$ that $$ ...