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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1answer
860 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 ...
4
votes
2answers
61 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
4
votes
2answers
141 views

$L^1$ norm of product of independent random variables

I am trying to show that $\|XY\|_1 = \|X\|_1\|Y\|_1$ for $X,Y$ independent random variables, where $\|X\|_1 = \int{|X| d\mathbb{P}}$. I have a feeling that this result is intuitive, but could anyone ...
4
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1answer
232 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
4
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1answer
2k 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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0answers
372 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
1k 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
69 views

“Empirical” entropy.

Information entropy is usually defined as $$\text{I}_b({\bf p}) = -\sum_{\forall i}p_i\log_b(p_i)$$ i.e. the expected value of the negative logarithm of the probabilities. This is all good when we ...
3
votes
1answer
96 views

Exercise on Martingales

I have been struggling with the following exercise and I was wondering whether my solution is correct or not. I am pretty sure about the second part of the question (the martingale part) but not so ...
3
votes
2answers
112 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
3
votes
1answer
77 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 ...
3
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0answers
109 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
votes
2answers
665 views

How can I show that the conditional expectation $E(X|X)=X$?

I tried to show that $E(X|X=x)=x$, which would lead me to get $E(X|X)=X$ but I am having trouble doing so. I know that the definition of conditional expectation (continuous case) is: ...
3
votes
1answer
134 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
450 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
132 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
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4answers
205 views

Conjunction fallacy [duplicate]

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
386 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
181 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 ...
3
votes
2answers
163 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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0answers
37 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
2
votes
2answers
385 views

Borel $\sigma$-Algebra definition.

Definition: The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ ...
2
votes
1answer
86 views

Is $(B_t^2)$ Markov where $(B_t)$ is Brownian motion?

I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not. Showing the square of a Markov process is or isn't Markov I guess I can repeat the method since to be Markov it ...
2
votes
1answer
103 views

Computing the expectation of conditional variance in 2 ways

Same as here. Let $X$ be a square integrable random variable on $(\Omega,\mathcal{F},P)$. Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Define the conditional variance of $X$ given ...
2
votes
2answers
164 views

At a party $n$ people toss their hats into a pile in a closet.$\dots$ [duplicate]

Question: At a party $n$ people toss their hats into a pile in a closet. The hats are mixed up, and each person selects one at random. What is the expected number of people who select their own hats? ...
2
votes
2answers
1k views

Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of X

Let X be a random variable with a continuous and strictly increasing c.d.f. function F (so that the quantile function F^−1 is well-defined). Define a new random variable Y by Y = F(X). Show that Y has a ...
2
votes
3answers
289 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
2
votes
5answers
276 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
958 views

How many times does a single fair die have to be rolled for a number to repeat

PHP developer here so no real math background :-( gave up after googling and hence the question As the question states how many times does a single die have to be rolled for any of the previously ...
2
votes
2answers
164 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
385 views

What is a real-valued random variable?

This question arose when someone (and surely not the least!) commented that something like $\left(X\mid Y=y\right)$ , i.e. $X$ under condition $Y=y$, where $X$ and $Y$ are real-valued random variables ...
2
votes
1answer
166 views

Question about Conditional Expectation

I have a seemingly trivial question regarding conditional expectation. Consider $x$ and $y$ be two integrable random variables on Probability space (X, $\Sigma$, $P$) such that $$E(X|Y) =_{a.s} Y$$ ...
2
votes
2answers
190 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
votes
2answers
45 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
383 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
1answer
3k views

Induction for sum of Poisson distributed random variables

Given the identically distributed and independant random variables $X_1,X_2,\ldots\sim\operatorname{Po}(\lambda)$ and $S_n=X_1+\ldots+X_n$ show with induction that ...
2
votes
1answer
519 views

Finite State Markov Chain Stationary Distribution

How does one show that any finite-state time homogenous Markov Chain has at least one stationary distribution in the sense of $\pi = \pi Q$ where $Q$ is the transition matrix and $\pi$ is the ...
2
votes
2answers
131 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
2answers
287 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 ...
1
vote
2answers
55 views

Maximum a Posteriori (MAP) Estimator of Exponential Random Variable with Uniform Prior

What would be the Maximum a Posteriori (MAP) estimator for $ \lambda $ for IID $ \left\{ {x}_{i} \right\}_{i = 1}^{N} $ where $ {x}_{i} \sim \exp \left( \lambda \right), \; \lambda \sim U \left[ ...
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3answers
49 views

Help with a Probability Proof

How do I prove the following: Show that if $A$, $B$ and $C$ are three events such that $P(A \cap B \cap C) \neq 0$ and $P(C | A \cap B)$ = $P(C|B)$, then $P(A| B \cap C) = P(A | B)$. Here is my ...
1
vote
1answer
52 views

Borel Cantelli lemmas or not?

I have this exercise "Let: $$X_1,X_2, \ldots , Y_1,Y_2, \ldots \sim U (0,1)$$ independent and identically distributed random variables. Is true that: $$\limsup \frac{X_n}{Y_n} = +\infty $$ almost ...
1
vote
1answer
40 views

The Continuity of Correlation Coefficient of a Continuous Stochastic Process

Given a continuous stochastic process with respect to time with finite variance at a given $t$. Does it necessarily imply, as $d\to 0$, 1) the covariance between $t$ and $t+d$ approaches the variance ...
1
vote
1answer
62 views

Show that the distance $D_c$ between densities is symmetric when the densities are related by a linear transformation

The distance between two density functions $p_0$ and $p_1$ is given by $$D_c(p_0,p_1)=\int_{p_0/p_1>c} (p_0-c p_1)\mathrm{d}\mu$$ where $c>1$ is a real number Question: Show that if ...
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vote
3answers
47 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} ...
1
vote
1answer
92 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
1
vote
0answers
58 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 ...
1
vote
1answer
140 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 ...
1
vote
1answer
86 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 ...
1
vote
1answer
58 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 ...