Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [probability-theory]

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 [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

1
vote
1answer
8 views

Find the conditional distribution of $X$ given that $Y=y$

Let $X$ and $Y$ be two random variable with density $f_{X,Y}(x,y)=\begin{cases} \frac{1}{y}, & \text{for } 0<x<y<1, \\[8pt] 0, & \text{otherwise}.\end{cases}$. To find the ...
0
votes
0answers
7 views

Multivariate Kolmogorov distance bounded by Wasserstein distance

I'm trying to find a bound for the multivariate Kolmogorov distance in terms of the Wasserstein distance. Denoting by $F$ and $G$ two cumulative distribution functions (cdf) on $\mathbb{R}^n$ the ...
1
vote
2answers
25 views

Let $\mu(X)=1$, $0 \leq f \leq k$, and $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$.

Let $\mu(X)=1$ for $\mu$ a positive measure. Let $0 \leq f \leq k$ for some $k\in\mathbb{R}$ and let $m=\int_X f d\mu$. Show $\int_X |f-m|^2 d\mu \leq \frac{k^2}{4}$. My attempt: I tried to expand ...
0
votes
1answer
23 views

Probability of a random subset of Z

I'm stuck in this question, could someone give me a hand? I'll post what I've done so far. Question 9: Let $A=(1,2,3,4)$ and $Z=(1,2,3,4,5,6,7,8,9,10)$, if a subset B of Z is selected by chance ...
0
votes
0answers
13 views

Entropy of max(0, uniform(-1, 1))

I'm trying to figure out how to deal with distributions that are mixtures of discrete and continuous. A simple example is max(0, uniform(-1, 1)) -- draw a (real) ...
0
votes
1answer
11 views

find the density function of $(U,V)$

Let $X=(X_1,X_2)^T$ be a random vector with 2-dimensional normal distribution, $E(X_1)=E(X_2)=0 , Var(X_1)=Var(X_2)=1$ and $Cov(X_1, X_2)= \nu$ with $|\nu| <1$. And let $Z \sim Bin(1,\alpha)$ be ...
1
vote
0answers
29 views

A lower bound on $|S_n - S_{n+1}|$

I am reading "Durrett: Theory and examples", fourth edition. In page 67, he proves theorem 2.3.7 (an application of Borel Cantelli lemma). During that demonstration, he arrives to a point where: $X_1,...
0
votes
1answer
12 views

Existence of random variable given infinite-dimensional probability measure

Let $\mathcal{X}$ be some possibly infinite-dimensional metric space and let $\mu$ be a Borel probability measure on $\mathcal{X}$. What theorem implies the existence of a probability space $(\Omega,...
0
votes
0answers
10 views

How to transform long term statistics into probability?

My concrete problem is the following: If the long term goal average for a soccer team is one scored goal every 60 minutes, what is the probability of scoring a goal in 60 minutes? What is the ...
0
votes
1answer
18 views

Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
1
vote
0answers
17 views

Sufficient statistics and natural parameters of exponential family

I am studying some properties of exponential family distributions, i.e., distributions whose pdf/pmf can be written (in its "natural" form) as $$f_X(\mathbf{x}\mid\boldsymbol \theta) = h(\mathbf{x}) \...
0
votes
0answers
27 views

Likelihood of getting flush, straight, etc

On Planet X, cards can take on a numerical value from $1$ to $7$ (inclusive) and their suit can be either red or blue. In a game of poker, each player gets three cards. 1) What is the probability of ...
1
vote
1answer
46 views

Sum of two random variables uniformly distributed on circles

Suppose we have two independent random variables $U_1$ and $U_2$ unfiorm on \begin{align} S_i = \left\{ (s_1,s_2) \in \mathbb{R}: \sqrt{s_1^2+s_2^2} =r_i \right\} \end{align} respectily. Assume $...
0
votes
2answers
34 views

Show that $\int_{-N}^N|x|^pF(dx)\le\limsup\int_{-N}^N|x|^pF_n(dx)\lt\infty$ [on hold]

I'm studying probability essentials question 18.16. $(X_n)_{n\ge1}$ have distribution functions $F_n$ and $X_n\rightarrow X$ in distribution. $p\gt0$. Show that for every positive N, $$\int_{-N}^N|x|^...
0
votes
0answers
16 views

How to aggregate the normal distributions of two Kalman populations?

Suppose I have the following Bayesian Network: It's given by the following probability distributions: $$\begin{aligned}X_1&\sim \mathcal N(\mu, \delta^2)\\ \forall i, 2\leq k\leq n: X_i|X_{i-1}&...
1
vote
1answer
18 views

Conditional Expectation with Respect to an Initial Condition

$\require{begingroup}\begingroup\newcommand{\dd}[1]{\,\mathrm{d}#1}$When studying diffusion processes, I often see the notion of expectation conditioned on an initial value. By this I mean the ...
0
votes
2answers
21 views

$\mathbb{E}(|X|^p)=\int_0^{\infty}pt^{p-1}\mathbb{P}(|X|\ge t)dt$

I found the following formula in a proof of Kinchin's inequality : $\mathbb{E}(|X|^p)=\int_0^{\infty}pt^{p-1}\mathbb{P}(|X|\ge t)dt\ \ p\ge 1$ where X is a finite sum of random independant variables $...
-1
votes
1answer
26 views

IF A,B,C are independent then prove that A and B ∪ C are independent [on hold]

IF A,B,C are independent then prove that A and B ∪ C are independent So, how can I prove it
2
votes
2answers
43 views

How could $E(X\mid Y)$ be a function of $Y$?

When I was solving $ \operatorname{Cov}(X,E(X\mid Y)) = \operatorname{var}(E(X\mid Y))$, I notice that $E(X\mid Y)$ was treated as a function of $Y$. My thinking is $E(X\mid Y)$ is taking values of $ ...
0
votes
0answers
13 views

Sow that one probability distribution can't be preferred to others.

Let $R= ${ $r_1, r_2, r_3, ...$} be a countable set of rewards, and let $U$ be a utility function on R. Let $P_1, P_2, P_3, ...$ be a sequence of probability distributions on R. For each distribution ...
2
votes
1answer
33 views

A new stopping time built from a stopping time

Let T be a stopping time for the filtration $(\mathcal{F_n})_{n \in \mathbb{N}}.$ For all $n \in \mathbb{N} \cup \left\{+\infty \right\},$ we set $\phi(n)=\inf\left\{k \in \mathbb{N};\left\{T=n \...
0
votes
2answers
32 views

coin/ sigma-algebra

You flip a coin two times. You consider two events: $$A=\{ " it \ lands\ heads \ up \ two \ times"\}$$ $$B=\{ " it \ lands\ tails\ up \ two \ times"\}$$ Which events do I have to add to get an ...
3
votes
2answers
57 views

Simple process in Itô calculus

For the definition of Itô integral, one uses simple stochastic processes. I have found two definitions for simple stochastic process, given a filtration $(\mathcal{F}_t)_{t\geq0}$, an interval $[0,T]$ ...
0
votes
1answer
50 views

Convergence of $\sum_{n=2}^{\infty}\left(\frac{\log n}{\log(\log n)}\right)^{-\log n/\log(\log n)}$

Initially, I need to prove, that $$\forall \lambda > 0 \ \forall\, \xi_i \sim \text{Pois}(\lambda), \xi_i\text{ are independent } \implies P\left(\limsup\limits_{n \rightarrow \infty}\frac{\...
0
votes
1answer
16 views

Tail bound for sum of random variables satisfying subgaussian upper tail bound

So suppose you have a collection of random variables $X_1, \cdots X_n$ that are iid and they all satisfy the tail bound $$ P(X_i-L>u)\leq \exp(-\frac{u^2}{2\sigma^2})$$ for all $u>0$. Is it ...
0
votes
0answers
7 views

Optimization over transition probabilities

I am interested in the following question: Fix a natural number $n\ge 2$. Let $\Omega=\{\omega_1, \omega_2, \dots, \omega_n\}$. A Markov process with state space $\Omega$ and transition matrix $P$ ...
0
votes
1answer
18 views

What is the covariance matrix of $4\times2$ data points? [on hold]

I do not know how to calculate the covariance matrix for a $4\times2$ data points The points are as followed: X | Y -2 | -1 -1 | 0 1 | 0 2 | 1 Can I get ...
0
votes
2answers
12 views

$(\frac1n \sum_{j\not=i} Z_j - \frac{n-1}nZ_i) \sim N(0, \frac{n-1}n)$

In lecture, we proved that $\overline{Z}$ and $\sum_{i=1}^n(Z_i-\overline{Z})^2$ are independent, but I don't follow one step. The proof is as follows. $Z_i \sim N(0,1)$ and $\overline{Z} \sim N(0,\...
0
votes
0answers
27 views

Convergence in probability of running maximum

Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
1
vote
1answer
24 views

Examples of martingales goes to $-\infty$

One example I can find is that set $X_{0}=0$, Let $X_{j}=\left\{\begin{array}{ll}{j^{2}} & {\text { with probability } \frac{1}{j^{2}}} \\ {\frac{-j^{2}}{j^{2}-1}} & {\text { with ...
0
votes
0answers
31 views

modeling of random variable

I have 2 random variables $U$, $V$ - uniformly distributed on $[0,1]$. How to model (simulate) random variable with distribution function $f(x) = \frac{1}{2}e^\frac{-x}{2} \mathbf{1}_{[0,\infty)}(x)$
1
vote
0answers
28 views

Random variable defined on the Lebesgue probability space

There is a random variable X defined on the Lebesgue probability space whose cumulative distribution function is F. We can find X(w) knowing that: $X(ω)=\inf\{x∈R:F(x)>ω\}$. 1) how do we prove ...
1
vote
0answers
29 views

Prove that $E[x | s, y] = b_0 + b_1 s + b_2 y$ for some constants $b_0$, $b_1$, and $b_2$.

Let $E[z | v]$ denote the conditional expectation of the random variable $z$ conditional on the random variable $v$. Assume that $s = x + \epsilon$, where $\epsilon$ is a random normal variable ...
0
votes
0answers
9 views

Convergence stopping time

If I have a sucession of continuous time stochastic processes, such that $L_t^N \xrightarrow{a.s} L_t$. Where $L_t^N$ is a jump process and $L_t$ is continuous (with respect to t). If $\tau^n = \inf\{...
0
votes
0answers
13 views

The relation between copula density and survival copula

I would appreciate if someone help me to solve the following problem. If I have copula $C(u_1,\ldots,u_n)$ it is obvious how to obtain copula density. It is $$c(u_1,\ldots,u_n)=\frac{\partial^n C(u_1,\...
1
vote
0answers
39 views

I don't understand the proof of Corollary 4.8.7 in the book of Ethier and Kurtz

I'm trying to understand the proof of Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas G. Kurtz. Here is the theorem and its proof:...
1
vote
1answer
18 views

Bounded, tight sets of measures are compact?

In Prokhorov's 1956 paper "Convergence of Random Processes..." it states the following. Where $\mathcal{R}$ is a complete, separable, metric space. Additionally, it says that any weakly convergent ...
0
votes
1answer
21 views

Matching problem - different answers

I found different answers for a problem I was trying to calculate. I will write it down, then write my solution and the other I found. Two players toss coins simultaneously until they have the ...
2
votes
0answers
29 views

A criterion of convergence almost surely.

Suppose random variables $\{X_k\}_{k\in \Bbb{N}}$ are $i.i.d.$ and set $S_n=X_1+...+X_n$, show that if $S_n/n\rightarrow 0$ in probability and $$S_{2^n}/2^n\rightarrow 0 \ \ a.s.$$ then $S_n/n\...
1
vote
0answers
22 views

Empirical distribution function converges against $N(0,1)$ in distribution

Let $(X_k)_{k \ge 1}$ be a family of i.i.d. random variables and $F$ be their distribution function. The empirical distribution function is defined as $F_n(t) = \frac{1}{n} \sum_{k=1}^n 1_{X_k \le t} ...
1
vote
0answers
25 views

Determine the error-probability of biased coin tosses using chernoff-bounds

Let's assume we have a biased coin with probabilities $\frac{4}{5}$ and $\frac{1}{5}$ and we don't know to which event (head or tail) the probabilities belong to. But we want to decide it by majority ...
1
vote
1answer
21 views

On a minimization problem involving the expectation of the logarithm of a normal random variable.

Let $X$ be a random variable distributed as a normal with variance $\theta \sigma^2$ I would like to show (I don't know if it's true) that $$E \left[ \frac{1}{2} \ln\beta X^2 - \frac{1}{2} \ln \theta ...
0
votes
1answer
27 views

Problem about a sequence of independent random variables with mean $0$ and variance $1$.

Let $X_n$ be a sequence of independent random variables with mean $0$ and variance $1$, show that for any bounded random variable $Y$, we have $\lim_{n\rightarrow \infty}\Bbb{E}(X_nY)=0$. I don't ...
1
vote
1answer
16 views

Does subgaussian tails imply the random variable has expectation 0?

If you have a random variable $X$ such that $$P(|X|>u)\leq C\exp(-\frac{u^2}{2\sigma^2})$$ holds for all $u>0$ can you conclude that $E(X)=0$. Likewise more generally if we have two separate ...
1
vote
0answers
17 views

Hoeffding's inequality generalization

Suppose I have independent random variables $X_1,\cdots ,X_n$ and a constant $L$ such that $$P(X_i-L>u)\leq C\exp(-\frac{ u^2} { 2\sigma_i^2})$$ Does Hoeffding's inequality apply here? I only ...
2
votes
3answers
102 views

Is it possible for an event $A$ to be independent from event $B$, but not the other way around?

I was wondering if event $A$ is independent of event $B$, would $B$ also be independent of event $A$? My original thought was that it should be independent, but then I realized if $A$ is independent ...
6
votes
1answer
101 views
+150

Understanding Optimal Transport in One Dimension.

I'm trying to understand these lecture notes. https://sites.ualberta.ca/~mathirl/IUSEP/IUSEP_2018/lecture_notes/Pass1.pdf I understand the formulation of the Monge Problem. However, I'm having trouble ...
0
votes
0answers
22 views

Conditional probability of events not arising from a well-defined function on S

Given a probability space $(S,\mathcal{F},P)$ where $S$ is countably infinite, $\mathcal{F}$ is the full $\sigma$-algebra, and $\forall a \in S, P(a) > 0$, is it reasonable to speak of $P(Y|X)$ ...
0
votes
1answer
50 views

Why does $P(E) < P(F)$ imply that $E \subseteq F$?

Why does $P(E) < P(F)$ mean that $E \subseteq F$ ? My reasoning (using Venn diagrams): It is seen clearly in the below picture that even if $P(E)<P(F)$, there is still some region in E that is ...
0
votes
0answers
18 views

I am having trouble figuring out how many lambda's (births) there are in a given birth-death Markov process problem.

These questions are not for assignment. I am just confused as to how to set up the problem. I also do not need help calculating the problems at hand. I understand that in a birth and death problem, $...