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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Does $X ⊥ Y \leftrightarrow X ⊥ Y | Z$ implies $(X,Y) ⊥ Z$?

Let $X, Y$ and $Z$ be random variables. Let $p_1$ be the statement that $(X,Y) ⊥ Z$ (meaning $(X,Y)$ and $Z$ are independent), $p_2$ be the statement that $X ⊥ Y$ (meaning $X$ and $Y$ are ...
0
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1answer
27 views

How to simplify this conditional expectation?

Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, is it correct that $$E_3[v_3\mid v_1 < \max\{v_2,v_3\}, ...
1
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1answer
27 views

apply the law of total expectation

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which ...
0
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0answers
24 views

Counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) \Rightarrow (Y_1 \perp Y_2) | X_1$?

Let $(Y_1 \perp Y_2) \mid (X_1, X_2)$ mean that random variables $Y_1$ and $Y_2$ are conditionally independent on $(X_1, X_2)$. Either is there a counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) ...
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1answer
35 views

Quick question on an example of bad probability theory.

From the text Probability with martingales by Williams. I don't understand why $p(C_n) = 0$, is it not $$P(C_n) = \lim_{n \rightarrow \infty} \# \{ k : 1 \le k \le n ; k \in C_n \} = \lim_{n ...
2
votes
3answers
45 views

Adding two discrete distributions

I am taking a probability course and I am having trouble adding two discrete distributions. The two distributions given are: $X$ has a discrete uniform distribution on the integers $0,1, ... ,9$. ...
2
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2answers
17 views

Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?

Let $X$ be a random variable defined on a measure space $(\Omega, F, P)$. Let $N$ be a sub sigma algebra of $F$. Then conditional expectation $E(X|N)$ is an a.e. equivalent class. Is the a.e. ...
2
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0answers
8 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
5
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4answers
56 views

Supermartingale vanishing at some stopping time

Let $\left\{X_t\right\}_{t\in[0, T]}$ be a continuous and non-negative supermartingale. We define the stopping time $$\tau_0:=\inf\{t\in[0,T]:X(t)=0\}\wedge T$$ and immediately obtain by continuity ...
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0answers
12 views

Calculating the Shannon information of drawing equal no. of cards

One card is drawn each from a $k$ deck of 52 cards where $k$ is a multiple of $52$. I need to prove that information of an outcome where each card appears the same number of times tends to ...
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0answers
13 views

exponential martingale [on hold]

If any one can help, I would really appreciate that. If I have random walk Sn constructed from summation of iid random variables Xi such that each Xi equals (Ai-Bi), where Ai is exponentially ...
0
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1answer
12 views

Bounding probability based on binary values

I've been reading this paper on probabilistic logic: http://ai.stanford.edu/~nilsson/OnlinePubs-Nils/PublishedPapers/problogic.pdf On page 76 theres a 3d diagram and Nilsson mentions the bounds on ...
1
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1answer
27 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
2
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1answer
12 views

Why $\Bbb{E}f$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$?

In my professor's lecture note there is a remark saying that "$\Bbb{E}[f]$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$". I think this should be easy, but I just don't see why. Can ...
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0answers
25 views

Can we simplify the conditional covariance $\mathbb{V}[(X\:Y\:Z)|X+Y+Z=1]$?

Given random variables $X,Y,Z$ on a probability space, can we write the conditional covariance matrix $$\mathbb{V}\left[ \left(\begin{array}{c}X\\Y\\Z\end{array}\right) \Bigg|X+Y+Z=1\right]$$ as a ...
2
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2answers
48 views

What does the conditional expectation look like when the $\sigma$-algebra is infinite

Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional ...
0
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1answer
30 views

Markov processes: Hitting times for a point form an i.i.d. sequence

Say that I have a recurrent time-homogenous diffusion process X (continuous strong markov process) and two points $x,y$. If $X_t$ goes from $y$, to $x$ and then back to $y$ again we denote it as a ...
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3answers
45 views

Random variable with 2 distribution functions

Just a question here, Given a random variable $X$ defined in a probability space, is it possible to have more than one distribution function $F$ ?
2
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1answer
28 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
0
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1answer
20 views

Terminology - “Sample space” vs “sample set”?

Given that a "sample space" is defined as the set of possible outcomes of a given random experiment, is there a fundamental reason to use the term "sample space" instead of "sample set" in probability ...
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0answers
27 views

Is continuous random variable always an “onto function” [on hold]

Is continuous random variable always an "onto function"? If yes, why?
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1answer
29 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
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1answer
23 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
2
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2answers
57 views

Is every $\sigma$-algebra generated by a partition?

I know that every finite $\sigma$-algebra is generated by a finite partition, but is every infinite $\sigma$-algebra also generated by "kind of" partition? Can anyone help provide a explanation or ...
3
votes
2answers
33 views

How to prove convergence in $L^p$ imply convergence in $L^r$ when $p>r$?

$X_n$ converges to $X$ in $p$th mean. Show that $X_n$ also converges to $X$ in $r$th mean when $p\ge r$. I have tried conditioning on $|X_n-X|\ge1$ and $|X_n-X|<1$ but no luck.
2
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0answers
39 views

How $\sigma$-algebra determines random variable?

In my probability textbook there is a statement saying that Knowing the $\sigma$-algebra $\sigma(X)$ generated by a random variable $X$ is equivalent to knowing $X$ itself. We equate $\sigma(X)$ ...
0
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1answer
27 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
6
votes
3answers
60 views

$\sum_{m=1}^{\infty }\frac{m}{2^n} \mu (E_{n,m}) \uparrow \int f d\mu$

Let $f \geq 0$ and $E_{m,n}=\{x :m/2^n \leq f(x) < (m+1)/2^n \}$ I need to show that as $n \uparrow \infty$ $\sum_{m=1}^{\infty }\frac{m}{2^n} \mu (E_{n,m}) \uparrow \int f d\mu$ My attempt: I ...
3
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1answer
76 views

Pointfree probability theory

I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like pointfree topology, where one basically replaces topological spaces by their locales ...
3
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1answer
86 views

Monty Hall problem again (from Grimmet and Stirzaker)

Grimmet and Stirzaker Exercise 1.4.5.2 In a game show you have to choose one of three doors. One conceals a car, 2 conceal goats. You choose a door but the door is not opened immediately. Instead ...
2
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2answers
36 views

Probability that two sets do not intersect

I'm trying to understand this simpler problem so I can apply the process to a more difficult homework problem. Let $U$ be a set with $n$ elements. Select $2$ independent random subsets $A_1, A_2 ...
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0answers
28 views

Understanding the set structure of probability theory [on hold]

Since events have their own probabilities and outcomes have their own probabilities. Why don't we just consider only one of events or outcomes directly? What's the motivation to have this set-point ...
2
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0answers
45 views

What's the sample space for a conditional expectation?

Define a probability space $(\Omega,\cal F,\Bbb P)$ and a $\cal F$ measurable random variable $X$, the conditional expectation given a sub $\sigma$-algebra $\cal F_0 \subseteq \cal F$ is a random ...
1
vote
1answer
41 views

How many ways can an integer $i$ appearing in a sequence with multiplicty at least $j$, be minimal

Let us construct an integer sequence of length $n$, where the integers are chosen from $\{1, 2, ..., k\}$, with i.i.d. uniform probability $\frac{1}{k}$. I want to compute the probability ($p_{ij}$) ...
2
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0answers
39 views

Is the sudden appearance of transient random walks in 3-dimensions a phase transition?

Consider a particle walking uniformly at random on the infinite d-dimensional lattice $\mathbb{Z}^d$. This is symmetric random walk. Symmetric random walk in two dimensions almost always returns to ...
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0answers
13 views

How to deduce this fact from the existence of factorized regular conditional probabilities and disintegration of probability measures?

In the lecture we had a theorem about the disintegration of probability measures in the following formulation: Theorem: Given two standard Borel spaces $(S_i,\mathscr S_i),i=1,2$ let $(S,\mathscr ...
2
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0answers
21 views

Uniqueness of the transformation turning random variables into IID uniform

We have two random variable $X:\Omega \to \mathbb R $ and $Y: \Omega \to \mathbb R^d, d \in \mathbb N$, $F_Y$ is the density function of $Y$ and $F_{X|Y=y}$ is a regular density function of $X$ ...
1
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1answer
56 views

Properties of independence and conditional independence

Recently, I see some properties from conditional independence wiki page https://en.wikipedia.org/wiki/Conditional_independence I don't quite understand the properties of "Rules of conditional ...
3
votes
1answer
20 views

Bernstein inequality for non-centered random variables (Is there a counterexample?)

The usual Bernstein inequality (see e.g. Rauhut + Fourcat, A Mathematical Introduction to Compressive Sensing, Theorem 7.30) states that if $X_1, \dots, X_m$ are independent mean zero random variables ...
0
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3answers
69 views

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable.

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable, because I could not make out one.
0
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1answer
37 views

hitting times and stopping times

stopping times are always hitting times, but not the other way around. As an example of this, Last exit times are not stopping times as they depend on future information. the last exit time of $A$: ...
2
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3answers
72 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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2answers
56 views

intuition of mass function of random variable [on hold]

When we are using $P\{X=x\}$ it seems like intuitively there is a function from $T$ (or measure from $\mathcal{B}(T)$) to $[0,1]$. What is the theoretical foundation behind this intuition?
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1answer
37 views

Find $E(X_1X_2 \mid X_2 X_3)$ for i.i.d. symmetric Bernoulli random variables $X_k$ [on hold]

I have problems with computing some basic conditional expected value - this is most likely don't via transformations of CEV, but I can't get them done properly. Can anyone help me with the following ...
1
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3answers
36 views

Definition of Random Variable on Measure Theory!

The definition is as following according to the book of John B. Walsh, Let $(\Omega, \mathbb{F}, P)$ be a probability space. A Random Variable is a real-valued function X on $\Omega$ such that for ...
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0answers
14 views

Why is the notion of an XRP, as opposed to an IID variable, useful in programming?

The most general notion which shares the main properties of i.i.d. variables are exchangeable random variables, introduced by Bruno de Finetti. Exchangeability means that while variables may not be ...
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0answers
28 views

Probability, expected frequency and resultant distribution skewed or not?

A population consisting of a certain proportion of defective items has mean $\mu = 2$. If a sample of 4 items is examined and repeated 200 times, obtain a) probability of an item being defective, ...
0
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1answer
27 views

Is there a Markov-type inequality for the Median?

Markov's theorem states that $P(|X| \geq a) \leq \frac{E[|X|]}{a}$. Is there an similar type of inequality that involves the median (somehow I doub't it, but I make no claim to comprehensive knowledge ...
2
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0answers
40 views

Conditional expectation of the maximum of two independent random variables, given one of them

The question I am interested in is the following: Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$, what is the conditional expectation of ...
0
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1answer
53 views

Why is $\int f+ g d \mu \geq \int f d \mu + \int g d \mu$ [on hold]

Hi I was studying probability theory and in extending the definition of integral from bounded functions with finite support to positive values functions I got stuck . As you can see in the picture I ...