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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2
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2answers
40 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
25 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
40 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
24 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 ...
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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
26 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
26 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 ...
0
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1answer
22 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?
1
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1answer
44 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
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2answers
31 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
36 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
26 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
55 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
votes
1answer
75 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 ...
2
votes
1answer
76 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
votes
2answers
27 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
votes
0answers
37 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
38 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
votes
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 ...
0
votes
0answers
12 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
19 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
52 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
66 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
35 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
votes
3answers
69 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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votes
2answers
55 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?
0
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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
vote
3answers
35 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 ...
0
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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
23 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
votes
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
votes
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
votes
1answer
51 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 ...
1
vote
1answer
45 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
-1
votes
1answer
56 views

How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
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0answers
15 views

Proving that a random variable is stable

If $\overline{F} = 1- F_X$ where $F_X$ is a cumulative distribution function of $X$. Then if the following is satisfied: $$ \lim_{x \to \infty} \frac{\overline{F}(\lambda x)}{\overline{F}( x)}= ...
3
votes
1answer
55 views

rolling a single die ten times

I have the following problem on a homework assignment for my Probability theory course: You roll a single six sided die ten times. What is the probability that you roll four 1's, three 2's, and three ...
0
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0answers
30 views

Conditional distribution

One point is chosen at random in the square $Q=\{|x| + |y| \leq 1\}$. Let $(X, Y)$ coordinates that point. a) The random variable $X$ and $Y$ are independent ? b) Find the density of $X$ given that ...
0
votes
2answers
29 views

Understanding the geometric distribution

Simple question that has to do with the interpretation of the geometric distribution and frequency function: $P (X=k) = (1-p)^{k-1}p $ for $k = 1,2,3... $ where we are interpreting X as being up to ...
2
votes
1answer
30 views

Probability that a stochastic process is below a special random level

Given a stochastic process $x(t)$ over time $t \in [0,T]$, and a given (deterministic) $\tau$, where $0<\tau<T$, define a random variable $x^{*}$ as $$ x^{*} \triangleq \inf\bigg\{y: ...
0
votes
1answer
18 views

Conditionally independent and intersection

I'm trying to show that, given events $A,B,C,D$, such that $A,B$ are conditionally independent given $C$, whether or not $A,B$ are conditionally independent given $C\cap D$. I spent a couple of hours ...
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0answers
21 views

Help with a definition involving multiple suprema/infima

I have trouble understanding a definition that comes up in a proof of the Prokhorov theorem. Let $E$ be a Polish space and $M$ a set of probability measures on the Borel $\sigma$-algebra on $E$. From ...
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0answers
24 views

Independence of Events in Lovasz Local Lemma

Let $G$ be a (finite) graph with maximum degree $d$ and vertices $v_{1}, \dotsc ,v_{n}$. Let us associate an event $A_i$ with $v_i (i = 1, . . . , n)$ and suppose that $A_i$ is independent of the ...
0
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2answers
48 views

Is it right to give equal chances?

I have a doubt ! Problem is : In a certain town, the probability that it will rain in the afternoon is known to be $0.6$ Moreover, meteorological data indicates that if the temperature at noon ...
0
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0answers
15 views

Laplace vs Fourier density representation of a positive rv

Given a general random variable $X$ with density function $f(x)$ and characteristic function $\phi_X(u)$ we can go back and forth between the density and the characteristic by using the Fourier ...
0
votes
1answer
20 views

A conditioned on B is independent of C

Let $A,B,C$ be measurable sets on a probability space. I'm trying to understand the meaning of the sentence: A conditioned on B is independent of C. The conditional probability was defined as: $$ ...
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0answers
25 views

Application of Riesz-Markov-Kakutani representation

Let A be finite set, $\{f_i\}_{i=1}^{n}:A\to \mathbb{R}$ non-negative and with the following property for every $\sum \lambda_{i}=1$: for any $g=\sum \lambda_{i}f_i$ there exists $a\in A$ s.t. ...
1
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0answers
37 views

Generating structure of Borel field

On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the author wrote: ...and there are Borel sets that cannot be arrived at from the intervals by any finite sequence of set-theoretic ...