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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36 views

expected value of expected value

I want to quantify the error of phase noise in terms of its normalized mean squared error. I define the error measure as (x is the error free function, y the distorted): $$ \rm NMSE = \frac{\int ...
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0answers
21 views

How to bound $E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right]$

I am looking for an upper bound on the following quantity \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] \end{align*} where $Z$ is ...
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1answer
31 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
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1answer
32 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
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2answers
58 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
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1answer
20 views

Indicator Functions with Random Variables

Let $E$ be an event and $Y$ a random variable. What exactly is meant by $\mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}]$? I have two guesses, the first is that $\mathbf 1_{Y\in B}$ is an indicator random ...
2
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1answer
37 views

Independence, conditioning, and correlation part 2 [closed]

Suppose $X$ and $Y$ are independent random variables. I now want to consider conditioning on some event $C$. Under what conditions will $X\mid C$ and $Y\mid C$ be correlated?
3
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0answers
26 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
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0answers
57 views

A weird problem on expected value of a random variable [closed]

Let $X$ be a discrete random variable taking values $x_1, x_2$, ... with probabilities $p_1, p_2$, ... respectively. Then the expected value of this random variable is $E(X)=\sum_{i=1}^{\infty }x_i ...
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1answer
18 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
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1answer
40 views

Equivalence of Conditional Expectations w.r.t. Discrete Random Variable

Let $X$ and $Y$ be integrable random variables such that $P(Y=y) > 0$ for all $y \in Y(\Omega)$. Then the conditional expectation of $X$ given $Y=y$ is defined as $$ \mathrm E[X \mid Y=y] : = ...
2
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0answers
26 views

Conditional Distribution absolutely continuous w.r.t Lebesgue measure?

Let $X,Y$ be integrable random variables. Then the condition expectation of $Y$ given $X = x$ is defined as $$ \mathrm E[Y \mid X=x] := \int_\Omega Y(\omega) \, P^X(\mathrm dw \mid x), $$ where ...
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1answer
36 views

Probability, drawing cards from 5 packs.

Firstly apologies for the vague post title. Apart from probability I don't really know what sub category this question falls within. If I have a pack of cards and I draw one card from it the chance ...
1
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1answer
41 views

Conditional Radon Nikodym

I am having some trouble conceptualizing and calculating a conditional RN derivative. When using this definition: I can see that if $\mathbb{Q} \ll \mathbb{P}$: $$\mathbb{E}_\mathbb{Q}(g) = \int ...
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0answers
20 views

Is the alias method “stable”?

The alias method is an algorithm for sampling from a discrete distribution. Let me describe it briefly. First there is a setup phase. You have $N$ values and associated probabilities. You introduce ...
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0answers
37 views

Convergence of argmax of poisson point process under a continuous map

This might just be a simple measure theory question, see the end remark, and apologies if that's the case. Here's the setup: Let $\mu_n$ be a sequence of Poisson point processes on a Euclidean space ...
1
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2answers
50 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
1
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1answer
28 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
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2answers
33 views

Markovian systems: Why must controls be independent of state?

I am currently working my way through Probabilistic Robotics by Thrun, Burgard, and Fox. On p. 91, I encountered the following statement: The Markovian assumption implies independence between ...
3
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3answers
38 views

Probability of Punctures for a group of cyclists

The matter of the probability of punctures occurring cropped up during a ride yesterday with a friend. His view is this, (As we can't let a subject drop.... ;-) ) "Eric, There must be more chance ...
3
votes
3answers
46 views

A sequence of random variables with bounded variance

If $\{X_n\}$ is a sequence of random variable with bounded variance: $$E|X_n^2|\le M<\infty,$$ and $X_n\to X$ in $L^1$, show that $$E|X^2|\le M.$$ I tried to use Fatou's lemma, ...
2
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1answer
61 views

Transformation theorem

Given $X_1$ is $\Gamma(\alpha,1)$ distributed and $X_2$ is $\Gamma(\beta,1)$ distributed and set $$Y=\frac{X_1}{X_1+X_2}.$$ The task is to show that $Y$ is $\operatorname{Beta}(\alpha,\beta)$ ...
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0answers
26 views

Instructive examples of independent sets on a probability space? [closed]

What are some good and simple examples of independent random variables / sets / sub-sigma algebras on a probability space? I am looking in particular for non-surprising, intuitive examples that ...
1
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1answer
53 views

Abstract enunciation of the Good Set Principle in measure theory

I am struggling with the Good Set Principle in Measure Theory, so is this rephrasing in the most abstract way ultimately correct? Good Set Principle Let $(X, \Sigma)$ be a measurable space. ...
3
votes
1answer
136 views

If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that k characters will be the same OR BETTER?

This is an addendum to a previous question found here. I have an alphabet: {A, B, C}. I'm randomly generating strings of length N from that alphabet. Examples: Examples: N=5, AACBC, AAAAA, BBCAA ...
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3answers
42 views

Intuition behind generating continuous random valiables

If we have a random variable $X$ with cumulative distribution function $F$ that is strictly ascending and we manage to find the inverse we can generate an instance $x_1$ from a uniformly distributed ...
0
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1answer
28 views

Asymptotic Inequality in Probability

Given that $P(X>a)\leq f(a)$. Now, $f(a)$ tends to zero faster than $P(Y>a)$. Does it mean that $(1)P(X>a) \leq P(Y>a)$ or $(2)P(X>a) \geq P(Y>a)$ as $a \rightarrow \infty$.
1
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1answer
32 views

Derivations in paper “Brownian Distance Covariance” and their intuition

I am reading a paper and I am stuck at following points. I tried to analyze but do not understand where to start. The paper "Brownian distance covariance" contains following lemma Notations: ...
6
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1answer
119 views

If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that exactly k characters will be the same?

I have an alphabet: {A, B, C}. I'm randomly generating strings of length N from that alphabet. Examples: Examples: N=5, AACBC, AAAAA, BBCAA What is the likelihood that exactly k characters of that ...
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votes
0answers
20 views

$\mathbb{F}$-Conditional density of a Brownian hitting time [closed]

Suppose that $(W_t)_{t\geq 0}\>$ is a standard Brownian motion on $(\Omega, (\mathcal{F}_t)_{t\geq 0 \>}, P)$ with $(\mathcal{F}_t)_{t\geq 0}\>$ the filtration generated by $W$. For $a\neq ...
0
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1answer
42 views

Current applications of the central limit theorem for binomial distributions

The central limit theorem in the binomal distribution case, also known as the De Moivre–Laplace theorem was historically used to approximate the binomal distribution with the normal distribution. I ...
3
votes
1answer
56 views

Is $E[Z E[Z^2\mid Y] ]$ positive or negative?

Let $Y=X+Z$ where $X$ and $Z$ are independent, zero mean, finite variance r.v. Moreover, $Z$ is Gaussian. Is there are way to say wether \begin{align*} E[Z \ E[Z^2 \mid Y] ] \end{align*} is positive ...
1
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0answers
33 views

Joint Distribution of k order statistics

For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest observation. I am wondering if anyone has ...
2
votes
1answer
43 views

“Empirical” entropy.

Information entropy is usually defined as $$\text{I}({\bf p}) = -\sum_{\forall i}p_i\log(p_i)$$ i.e. the expected value of the negative logarithm of the probabilities. This is all good when we have ...
2
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0answers
30 views

Potential measure of the product of (independent $\alpha$-stable) subordinators

For a nondecreasing Levy process $\mathbf{X}$ with values in $[0,\infty)$ (i.e. a subordinator) Jean Bertoin defines the potential measure of $\mathbf{X}$ in his book "Levy processes" as follows (p. ...
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0answers
33 views

difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized ...
0
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1answer
21 views

Dividing by epsilon and taking the limit of epsilon to zero

In deriving the joint distribution of two order statistics, there is the following step (F(x) is the Cumulative Dist Function at x, f(x) is the PDF at x): $$ ...
2
votes
1answer
22 views

Necessary likelihood ratios conditions for stochastic dominance

Suppose $X$ has CDF and PDF $F_X$ and $f_X$ with support $(-\infty,\infty)$ and $Y$ has CDF and PDF $F_Y$ and $f_Y$ also with support $(-\infty,\infty)$. Muller (2001) claims that if $X$ is ...
0
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2answers
49 views

If we know X is a Poisson binomial random variable what can we say about mX?

Suppose that X is sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has Poisson binomial distribution. Is mX also a Poisson binomial random ...
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2answers
46 views

Tricky Markov Chain

I found this problem a bit tough and was wondering if you could give it a go (especially the last part). This goes as follow : A gambler wins $1$ dollar at each round, with probability $p$, and ...
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1answer
86 views

How can I prove if $Y\leq X$ then $E[Y]\leq E[X]$?

If $Y\leq X$ always holds, then $E[Y]\leq E[X]$. How can I prove this (formally)? Also, can the equality happen if we know that $Y=X$ does not always hold? (i.e. $X$ and $Y$ are not exactly the same)
6
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2answers
110 views

Generalized convex combination over a Banach space

The Question: Is the following true? If not, what further hypotheses do I need? Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, ...
1
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1answer
40 views

Computing $E[ {\rm Tr}\{(ZZ^T)^2 \}]$ for $Z$ Gaussian.

Let $Z \in \mathbb{R}^n$ be a Gaussian random vector with zero mean and $Cov(Z)=I$ where $I$ is identity matrix. How to compute \begin{align*} E[ {\rm Tr}\{(ZZ^T)^2 \}] \end{align*} I know that ...
0
votes
1answer
47 views

Is the Expected Value fundamentally a theory(em) or a mere definition\specific measure? [closed]

At first, I was presented with the Expected Value as a definition. It is also a noun, and it is defined to be the implement of a specific formula. Now, I know it is a mathematical measure that is used ...
0
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1answer
20 views

Probabllility of a single trial given the probability of two random trials..

I came across a statement while reading an article. Please explain me these statements Pr[y1,y2] [f(x+y1) - f(y1) = f(x+y2) - f(y2)] > 5/9 Since we have ...
0
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0answers
26 views

Independence of random variables after a transformation

We are given four random variables $S:=(S_1,S_2,S_3,S_4)$ defined on $(\Omega, \mathcal A,P)$. The random variables can be viewed as being extracted from a stochastic process. I assume that all ...
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3answers
97 views

How is it possible for two random variables to have same distribution function but not same probability for every event?

It is completely out of the world for me to hear that such a case exists. I was shocked and could not develop any intuition as to how it is possible. It also breaks my understanding (intuitive) of the ...
0
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2answers
40 views

What happens when we take a compliment in probability and why is sigma algebra needed?

When we take complement of a set, do we mean sigma algebra minus the set or only the sample space minus the set. Also why is sigma algebra needed in the axioms of probability ? For reference the ...
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1answer
39 views

almost sure Limit of independent events [closed]

Problem: Let $A_n$ be independent events with $P(A_n)=\frac{1}{n}$. Let $Y_{k+1}:=min\{n>Y_k: A_n\ occurs\}$, $Y_0=0$. Show: $\lim \inf_{n\to\infty}Y_{k+1}-Y_k \ge 2$ almost surely Hint: ...
-1
votes
2answers
53 views

almost sure convergence given density [closed]

my problem: Let $X_n$ be iid random variables with density $f(x)=\frac{1}{2}x^{-2}1_{\{|x|>1\}}$. Show that $\frac{X_n}{n}$ does NOT converge almost surely. Can anybody help me?