Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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262 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
2
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2answers
769 views

Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = ...
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1answer
960 views

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

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
2
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1answer
109 views

A question about a stochastic process being Markov

Let $(X_{s},\mathcal{F}_{s})$ be a stochastic process adapted to a given filtration. I was told that, in order to prove that $X$ is Markov, it suffice to prove that for any nonnegative, ...
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5answers
2k views

Some case when the central limit theorem fails

If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite ...
2
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1answer
994 views

Rule with independent random variables and conditional expectations

I want to use a rule for conditional expectation I found in (German) wikipedia, not in my script/textbook of probability theory, I guess it should be simple and follow more or less straight from the ...
2
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0answers
2k views

Expected value of max/min of random variables

I am trying to solve the following problem. Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
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3answers
850 views

Is expectation Riemann-/Lebesgue–Stieltjes integral?

In probability theory, when having $ E(f(X))=\int_{-\infty}^\infty f(x)\, dg(x) $, an expectation of a measurable function $f$ of a random variable $X$ with respect to its cumulative distribution ...
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1answer
36 views

Conditional mean on uncorrelated stochastic variable 2

This question is a follow up from this. I was in doubt if to add it in my previous question, but thought it unfair to the great answers I had received. Let $X,Y$ be stochastic variables such that ...
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2answers
149 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
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1answer
475 views

Prove that vector has normal distribution

You are given two independent random variables: $W \sim \mathrm{Exp}(1)$, $Q \sim U([0; 2\pi ])$. Also, $a$ is a constant, chosen from $[-\pi/2; \pi/2]$. You build following random variables, based ...
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1answer
54 views

One confusion over conditional expectation

Suppose for two random variables $X_1$ and $X_2$ $\sigma(X_1) = \sigma(X_2)$. Why $E[Y | X_1] = E[Y | X_2]$ a.e. ? the set where $X_1= X_1(\omega)$ can be different from the set $X_2= X_2(\omega) $. ...
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votes
1answer
292 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
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votes
2answers
351 views

Independence of disjoint events

I'm taking a class in Probability Theory, and I was asked this question in class today: Given disjoint events $A$ and $B$ for which $$ P(A)>0\\ P(B)>0 $$ Can $A$ and $B$ be independent? ...
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1answer
1k views

Asymptotics of binomial coefficients and the entropy function

I found a question while I was trying to practice Combinatorics and Probabilistic methods.I tried to solve it with no success.. this is the question: Use the Stirling approximation of the ...
6
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2answers
165 views

$X$ and $f(X)$ independent $\Longleftrightarrow$ $f(X)$ is degenerate

Let $(\Omega, \cal{A}, \mathbb{P})$ be a probability space and $X$ a random variable on $\Omega$. Let, also, $f:\Omega\to\mathbb{R}$ be a Borel function. Then: $X$ and $f(X)$ are independent ...
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0answers
184 views

Regular Version of Conditional Gaussian Distribution

Let $Z_{1}$ and $Z_{2}$ be two independent normally distributed random variables with expectations $\mu_{1},\mu_{2}\in\mathbb{R}$ and variances $\sigma_{1}^2,\sigma_{2}^2\in (0,\infty)$ . I would ...
4
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1answer
167 views

central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is ...
4
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1answer
972 views

Why does a time-homogeneous Markov process possess the Markov property?

Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
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votes
5answers
430 views

Sleeping Mathematician (Sleeping Beauty)

I came across the following thought experiment, and I would like to understand whether the controversy around it is justified. Imagine an experiment in which a mathematician is put to sleep with some ...
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1answer
664 views

Relations between Order Statistics of Uniform RVs and Exponential RVs

Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
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0answers
292 views

Inequalities involving the probability density function and variance

I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
4
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2answers
773 views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
3
votes
1answer
71 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
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0answers
90 views

Random $0-1$ matrices

I'm working my way through the Oxford notes in Probabilistic Combinatorics and came across this question in one of the question sheets; I'd like to stress that this is not my homework: I'm simply ...
3
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1answer
114 views

How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?

I not solve the follow limit $$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$ where $(\Omega, \mathcal{F}, \mu)$ is a probability ...
3
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1answer
374 views

Bus stop probability question

People arrive at random times and independently at a bus stop and wait for the bus to arrive. The bus arrives at this stop once every hour. Thus, the waiting times of the people follow a uniform ...
3
votes
1answer
115 views

Product Measures

Consider the case $\Omega = \mathbb R^6 , F= B(\mathbb R^6)$ Then the projections $\ X_i(\omega) = x_i ,[ \omega=(x_1,x_2,\ldots,x_6) \in \Omega $ are random variables $i=1,\ldots,6$. Fix $\ S_n = ...
3
votes
4answers
164 views

Conjunction fallacy

I was reading this article which has the following question, Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with ...
3
votes
2answers
370 views

Expectation of a stopping time uniquely determined by a function

Let $(X_t)_{t\ge0}$ be a Markov chain on a finite state space $\Omega$, with transition probability $P$. Let $T$ be a stopping time such that $T=\min \{t\ge 0;X_t \in A \subset \Omega \}$.  If ...
3
votes
1answer
163 views

Relation: pairwise and mutually

Suppose we can define a relation $R$ over the sets $X_1, …, X_k$ for any natural number $k$, note not specified for a particular $k$. I was wondering if there is some definition or conditions ...
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2answers
153 views

probability -Diverging expectation

As I keep reading probability books, there are always some issues that no one considers. For example, for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega ...
2
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2answers
126 views

conditional expectation of brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$. It is intuitive that, for fixed $s<t<u$ $$\mathbb{E}[B_t\mid \sigma(B_s,B_u)]=B_s+\frac{t-s}{u-s}(B_u-B_s).$$ However, I ...
2
votes
5answers
247 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
2
votes
2answers
118 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
2
votes
2answers
158 views

Other way to express $e^{|x|+|y|}$

I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...
2
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2answers
41 views

$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y$

$X \sim \mathrm{Unif}[0,1], Y|X \sim \mathrm{Unif}[0,X^2].$ Find PDF of $Y.$ Solution. $$f_{Y|X}(y|x) = \frac{1}{x^2}, \text{ $x \in (0,1]$, $y \in \mathbb{R}$.}$$ Thus $$f_{X,Y}(x,y) = ...
2
votes
2answers
243 views

Distribution of sums

I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that ...
2
votes
2answers
125 views

sub martingales and more

This is a problem on sub-martingales. Given : $X_n = X_0 \mathrm{e}^{\mu S_n}$, $n= 1,2,3,\ldots$, where $X_0 > 0$ and where $S_n$ is a symmetric random walk and $\mu$ is greater than zero. We ...
2
votes
3answers
402 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
2
votes
2answers
255 views

Detail in Conditional expectation on more than one random variable

I have $E(X|Y,Z)=0$, $X$ independent of $Y$ and of $Z$ and I want to conclude that $E(X)=0$ ($X,Y,Z$ are real-valued random variables). Okay it seems quite obvious, but if I try to make a strict ...
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4answers
2k views

Why isn't there a uniform probability distribution over the positive real numbers?

Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what ...
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0answers
34 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
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1answer
98 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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1answer
76 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
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1answer
55 views

Help with a problem involving limit theorems

Here's a problem I am stuck on. The problem goes as follows: Suppose the distribution of scores of a test has mean 100 and standard deviation 16. Calculate an upper bound for the probability ...
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0answers
133 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
51 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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1answer
33 views

Given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, prove that $P(C\mid A)≥1-\varepsilon$

We need to show that, given $P(B\mid A)=1-\varepsilon$ for some $0<\varepsilon<1$ and $P(C\mid B)=1$, that $P(C\mid A)≥1-\varepsilon$. Since we know that $P(C\mid B)=1$, it follows that $P(B ...
1
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1answer
68 views

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds:

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds: $P(Z\leq -u \space \text{or} \space Z\geq v) \leq \frac{4\sigma^{2} + ...