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 [tag:probability-...

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53 views

Extending a probability measure to the sigma field obtained by adjoining a new set

Suppose that $P$ is a probability measure on a sigma field $\mathcal{B}$ and suppose $A\not\in\mathcal{B}$. Let $$\mathcal{B}_{1}=\sigma(\mathcal{B},A)$$ be the sigma algebra generated by $\mathcal{B}...
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1answer
16 views

Convergence of time translated iid process

Asume we have a sequence of i.i.d. rv $(Y_n)_{n\geq 0}$, with finite expectation. If $\sqrt{n}^{-1}Y_n\rightarrow 0$ almost sureley, can one conclude, that $\sqrt{n}^{-1}Y_{n+m}\rightarrow 0$ almost ...
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0answers
25 views

Calculate conditional expectation using formal definition

Let $\xi$ and $\eta$ be two random variables uniformly distributed on $[0, 1]$. What is conditional expectation $\Bbb{E}[\xi|\xi+\eta=1]$? I know that i can get it using symmetry: $$\Bbb{E}[\xi|\xi+\...
1
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0answers
33 views

Independent random variable in the limit (need extension?)

Given a sequence of rv $(X_n)$ on $(\Omega,\mathcal{F},P)$ with values on $(E,\mathcal{E})$ where $\mathcal{F}=\bigcup \mathcal{F}_n$ and $\mathcal{F}_n=\sigma(X_s:s\leq n)$. Lets say there is a ...
0
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0answers
9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...
0
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1answer
108 views

Average number of terms required for a sum of exponential variables to reach a specific limit

I have a sum $\sum_{i=1}^\infty Y_i$ where $Y_i=AX_i+a$ if $X_i>X_{lower}$ and $Y_i=BX_i+a$ if $X_i<X_{lower}$. Here $X_{lower}, A, a, B$ are positive constants and all $X_i$'s are i.i.d ...
1
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1answer
31 views

Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
0
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1answer
31 views

Interpretation of inequality between 2 stopping times.

Let $\mathcal{F}_{n}$ be a filtration of $\sigma$-fields, i.e., $\mathcal{F}_{n} \subset \mathcal{F}_{n+1}$ for every $n \ge$ 1. Let $\tau_{1}$ be a stopping time, i.e, a random variable such that the ...
1
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1answer
33 views

Finding the pdf of the difference of minimum and maximum of a finite set of random variables.

Let $X_i$ $ (1\leq i\leq n)$ be identically distributed uniformly on $(0,1)$. Let $U = \min_i(X_i)$, $V = \max_i(X_i)$. Find the pdf of $V-U$ This is what I did. I found the cdf and differentiated. ...
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0answers
38 views

Distribution of sum of two random variables not necessarily independent

Let $X$ and $Y$ be two random variables, not necessary independent. How can one obtain $P(X+Y|X)$? I know how to use convolution to find $P(X+Y|X)$ when $X$ and $Y$ are independent.
2
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1answer
29 views

Correlation between lagged Brownian motions.

Say I have two Brownian motions $X^1$ and $X^2$. Say they have constant correlation $\rho$. Then of course I know the correlation between $X^1_t$ and $X^2_t$. Furthermore I know that correlation ...
0
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1answer
48 views

Distribution Convergence of an Random Variable

I need to show that $$\frac{\sqrt{2n}}{\theta +1}\left(\frac{1}{\bar{X}_n}-1-\theta \right) \to^{d} N(0,1)$$ where $\bar{X}_n = \frac{1}{n} \sum_{i=1}^nX_i$ and iid random variables $X_i$, $X_1 \...
5
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0answers
135 views

Demystifying invariant measures in probability theory

Just trying to understand, at least conceptually, invariant measures, and specially their role in probability theory. To be brief: I understand that if we have a set $X,$ with $A \subset X$ just a ...
3
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2answers
52 views

Determine whether a random binary sequence was generated by human or natural process

Given a binary sequence, how can I calculate the quality of the randomness? Following the discovery that Humans cannot consciously generate random numbers sequences, I came across an interesting ...
1
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0answers
19 views

Does the binomial random graph model $G(n, (\ln n)/n^2)$ obey zero-one law?

I want to know if the bionomial random graph model $G\left(n, \frac{\ln n}{n^2}\right)$ obeys Zero-one law or not? I know that $\frac{\ln n}{n}$ is a threshold function for connectivity and for $\...
1
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1answer
27 views

Probability of G(4, 1/2) random graph being connected

Suppose we have a random undirected graph G(4, 1/2), i.e. the probability of any two of the four total vertices being connected is 1/2. how to find the probability that Probability (G(4, 1/2) is ...
6
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0answers
151 views

Limit distributions for Markov chains $X\to\sqrt{U+X}$

This question spawned from a recent, very interesting problem. Let $\varphi=\frac{1+\sqrt{5}}{2}$ and $T$ denote the map on the space of continuous probability density functions supported over $\...
1
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1answer
19 views

On a condition for a.s. finite stopping times

Assume that $\{\tau_n: n \in \mathbb{N}\} $ is a sequence of stopping times with respect to some filtration such that $P[\tau_n < \infty] = 1. $ Is that true that there must exist a sequence of ...
3
votes
1answer
27 views

When can a set of numbers be the moments of a random variable?

Suppose that I have a set of known measurable scalar-valued functions $f_{1},\ldots,f_{K}$. Associated with these functions, I also have a set of known real numbers $a_{1},\ldots,a_{K}$. Under what ...
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0answers
33 views

Geometric mean of random geometric variables converging with probability to a constant

I have the following question at hand.. Let $X_1,X_2,\cdots, X_n$ be a sequence of iid random variables with common uniform distribution on $[0,1]$. Define $$Z_n=\left(\prod_{\ i=1}^{\ n}X_i \...
7
votes
1answer
216 views

How to prove that there exists $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions. We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
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1answer
40 views

Probability of breaking a string at random point [closed]

A string of length $1\operatorname{m}$ is to be cut at a random point. What is the probability that the longer part of the string is at least twice longer than the shorter part. It means that the ...
0
votes
1answer
14 views

Intuitive definition of scaling random variables by a constant?

From how I understand scaling discrete random variables, we are multiplying all members in the set by the scaling constant. I.E if our random variable X = {1,2,3,4} and our scaling factor is $\alpha ...
1
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2answers
42 views

How does the infinite union work in $\sigma$-algebra?

Reading a probability theory book, and it says that if we have a sample space $\Omega$, then some class $F$ of subsets of $\Omega$ makes a $\sigma$-algebra and has the following properties: 1) $\...
1
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1answer
21 views

Scaling Random variables by a constant

First, let me state the problem: "Customers at Fred's cafe win a 100 dollar prize if their cash receipts show a star on each of the five consecutive days Monday...Friday in any one week. The cash ...
1
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1answer
18 views

PGF of sum of $N$ random variables, where $N$ is a random variable itself.

Let's say $X_i$ (for $i = 1, 2, \ldots$) are independent r.v.'s that return $0$ or $1$, both with probability 0.5. Let's say $N$ is a geometric random variable with $P(N=n) = 0.5^n$ for $n=1, 2, \...
1
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0answers
16 views

Geometric interpretation of multiplication of probabilities?

When dealing with abstract probability space $\Omega$ which consists of atomic events with measure ($P: \Omega \rightarrow \mathbb{R}$) defined for them, it seems natural to start immediately ...
1
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2answers
41 views

Picking cards sequentially vs consecutively

We have a pack of 6 cards over the table. Cards are: {A, A}, {B, B}, {C}, {D}. There are 3 players (Papa, Pepe, Popo) sat around the table. Cards are all upside down, so the players cannot see which ...
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0answers
27 views

Show that $E(\mathrm{var}(Y|X)) \leq (1 - \mathrm{corr}(X,Y)^2) \mathrm{var}(Y)$

Expectation of conditional variance (Exercise 4.6.7 from Grimmett and Stirzaker): Let $X$ and $Y$ be random variables with correlation $\rho$. Show that $E(\mathrm{var}(Y|X)) \leq (1 - \rho^2) \mathrm{...
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1answer
35 views

Doob Decomposition Theorem - submartingale iff increasing

Probability with Martingales To prove $b$ I tried: $$A_n \ge A_{n-1}$$ $$\iff E[X_{n} - X_{n-1} | \mathscr F_{n-1}] \ge 0$$ $$\iff E[X_{n} | \mathscr F_{n-1}] \ge X_{n-1}$$ That ...
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1answer
23 views

Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Probability with Martingales This is my understanding of what is going on in the proof above: We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in ...
0
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1answer
105 views

Almost sure convergence of equal weighted sum

Let $Z_1, Z_2, ...$ be independent random variables in the same probability space defined as follows: $$P(Z_n=n)=P(Z_n=-n)=\frac{1}{2n^2} \space \mathrm{and} \space P(Z_n=0)=1-\frac{1}{n^2}$$ Is it ...
0
votes
1answer
31 views

Do independent events always come in clusters?

I was reading a website that reports occurrences in commercial aviation. Recently there have been a number of similar incidents involving tyre problems (tyre damage, blown tyres, etc) on different ...
1
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0answers
17 views

Variance as a sum of conditional variances

So, for a random variable $X$ and an event $A$, the following expression of the mean value of $X$ as a sum of conditional expectations is valid, or at least, i could prove it for discrete random ...
0
votes
1answer
22 views

When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is ...
3
votes
1answer
74 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
0
votes
4answers
46 views

Two dice are rolled and the sum of the face values is less than six. What is the probability that at least one of the dice came up a three?

Two dice are rolled and the sum of the face values is less than six. What is the probability that at least one of the dice came up a three? This is just taking the initial question that I asked and ...
1
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2answers
91 views

Probability Question on Speed Dating

Suppose I have 2 groups of 20 people. 20 Male and 20 Female. After doing some speed dating, each Man writes down 2 women he fancies. And Each women writes down 2 men she fancies. Everybody has a ...
2
votes
2answers
88 views

Cholesky decomposition of a covariance matrix with swapped order of variables

Could you please let me know if there is a quick way to recompute result of a Cholesky decomposition of a covariance matrix, if the order of variables was switched to put a different variable as #1 on ...
5
votes
2answers
65 views

Average shortest distance between a circle and a random point lying in it

What is the average shortest distance between the circle $(x-a)^2+(y-b)^2=r^2$ and a random point lying in it? This question is just idle curiosity. Basically, it's the same as finding the ...
2
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0answers
22 views

Probability that there exists $M>0$ such that two processes $\{X_t\}$ and $\{Y_t\}$ are smaller than $M$ at the same time, for infinitely many $t$.

Suppose I have two iid sequences of random variables $\{X_t\}_{t\in\mathbb{N}}$ and $\{Y_t\}_{t\in\mathbb{N}}$, both absolutely continuous with full support. I know that with probability one, for any $...
0
votes
0answers
34 views

Is distribution of $Y = \sum_{n=1}^{\infty}0.5^{n} X_n$ Lebesgue measure? [closed]

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent binary random variables defined over probability space $(\Omega,\mathcal{A},P)$ such that $P(X_n = 0) = P(X_n = 1) = 0.5$. Define $Y = \sum_{n=...
1
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0answers
93 views

Probability of losing lotteries needed

A person earns $x_i$ amount of money every month where $x_i$ is an exponential random variable with parameter $\lambda_1$. The amount $x_i(1-p)y$, here $0 \leq p\leq 1$ and $y$ is exponentially ...
1
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0answers
26 views

Conditional Independence of Correlated Stochastic Processes

Let $\{X_1(t)\}$ and $\{X_2(t)\}$ be discrete-time stochastic processes, such that $X_1(t) = f(I_{t-1},\{Y_{1 t}(I_{t-1})\},\{X_{1 \tau}\}_{\tau \leq t-1} ),$ and $X_2(t) = g(I_{t-1},\{Y_{2 t}(I_{t-1})...
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1answer
45 views

What does it mean for a function to be nonrandom? [closed]

Given a probability space, I am guessing that a random function is a function of a random variable. As such a random function is measurable under the probability space. What does it mean for a ...
0
votes
2answers
28 views

Random variables and independence of $\sigma$-algebras

If a random variable $X$ is independent from the $\sigma$-algebra $F_t$ for every $t$ in a collection of indexes, is it true that $X$ in independent from the $\sigma$-algebra generated by all the $F_t$...
0
votes
1answer
26 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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0answers
22 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
1
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0answers
37 views

$\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ - definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
1
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0answers
47 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...