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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
19 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 ...
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 ...
0
votes
2answers
51 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 ...
4
votes
0answers
43 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
26 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
19 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 ...
-3
votes
0answers
37 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 ...
-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
1answer
59 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 ...
0
votes
0answers
54 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} ...
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
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 ...
2
votes
0answers
27 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 ...
-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
1answer
44 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 ...
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
1answer
38 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 ...
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 ...
3
votes
0answers
42 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 ...
0
votes
2answers
25 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 ...
1
vote
0answers
28 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
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 ...
8
votes
0answers
34 views
+100

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 ...
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 ...
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
1answer
24 views

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

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, ...
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 ...
0
votes
0answers
21 views

Minimum Random Variables and Integration

We are given a sequence of independent random variables $\lbrace X_{nk} \rbrace$, for $k=1,...,r_{n}$, with $E(X_{nk})=0$ and $\sigma^{2}_{nk}<\infty$. My question involves a small piece of the ...
0
votes
0answers
14 views

Kullback-Leibler Divergence (KL) and Approximation Symmetry Property

The Kullback-Leibler Divergence doesn't satisfy the symmetric property. But, it can be approximated (bounded) to such a value. in this paper: Compressing Interactive Communication under product ...
3
votes
1answer
55 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
0answers
35 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
30 views

Problems deriving probability generating function for the negative binomial distribution [on hold]

My problem is the following: Part A.a I can't get the moment generating function to be what it states in the exercise. And I found other people asking the same question but they get a different ...
0
votes
0answers
7 views

optimization technique to find the best result

i have two outcomes from two types of test. both the results are not 100% accurate. Is there any technique available to extract the final result from these two outcomes?
0
votes
1answer
23 views

Inequality for Expected Value of a Convex Function

Let $X$ be random variable with $-1 < X < X_\max$ and $X_{\max} >0$ and take $a = \min\{1, \frac{1}{X_{max} }\}$, I was wondering is the following inequality hold $$ E\left[ {\frac{X}{{1 ...
0
votes
1answer
17 views

Let W and Z be two random variables such that W ≤ Z. Show that for any ε > 0, P(W > ε) ≤ P(Z > ε).

I feel like I should be able to use Markov's Inequality, but have not found an effective way to use it for this problem. Any help would be appreciated.
2
votes
1answer
35 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
11
votes
3answers
344 views

Help understanding the weak law of large numbers with respect to statistics

I'm trying to do some self-studying to upgrade my statistics knowledge, and came across this term in a section discussing the weak law of large numbers and Bernoulli's theorem: $$\sum_{k=0}^n ...
1
vote
1answer
62 views

Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
0
votes
1answer
12 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
4
votes
1answer
62 views

$\lim_{n \to\infty} E(|S_n|)= \infty$ for $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1, E(X_1)=0$

Problem: For $(X_n)_{n \geq 1}$ i.i.d. real RV with Var$(X_1)=1$ and $E(X_1)=0$ and $S_n$ denoting the partial sum of the RVs we have $$\lim_{n \to \infty} E(|S_n|)=\infty $$ My Approach: I ...
0
votes
0answers
28 views

What is the meaning of probability of an edge connected by two nodes in a graph

I am studying random graph models. While studying random graph models if we want to generate for instance erdos renyi's random graph model then we will have to place $n$ vertices and connect each pair ...
0
votes
0answers
48 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = ...
1
vote
0answers
23 views

Esscher-Transform/ Levy-Process: Measure induced by trajectory

For a Levy-process $X_t$ w.r.t. to a measure P we define $\Theta$ as the set, for which $E[exp(\theta X_t)]$ is defined and finite. Note $\Theta$ is independent of $X_t$. Define ...
2
votes
0answers
52 views

How to Find Expected Value from a Joint Distribution?

I am trying to solve the following problem: Let $X$ be a random variable from a contaminated normal distribution. That is, let $B ∼\text{Bernoulli}(p).$ Then $X|B = 0 ∼ N(µ, τ^2 )$ and $X|B = 1 ...
1
vote
0answers
38 views

Basic measure theoretic definitions of random variables/probability distributions: codomain versus range?

I have the following question: in the collection of all measure theoretic definitions of random variables/probability distributions/cdf/pdf it seems to me that what is considered is always the ...
0
votes
0answers
48 views

Measure theoretic definitions of random variables/probability distributions

I'm asking your help to answer questions (a),(b),(c) outlined in the summary below. The questions are so connected that I found difficult to ask them separately. Could you also let me know if you find ...
1
vote
0answers
27 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
1
vote
1answer
26 views

Equality of Measures and Intuition

Caveat: I have no formal training in measure theory and am learning as I go. The concept in this question is puzzling me: Equivalent measures if integral of $C_b$ functions is equal I'll re-state ...