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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39 views

Forming a triangle probabilistically [duplicate]

What is the probability that if you break a stick at $2$ points the three sides form a triangle? Is there a technique that avoids calculus?
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0answers
27 views

Representation of point process

Let $E$ be a polish space and $N(E)$ be the space of finite integer value measures. It is known that for every $\mu \in N(E)$ exists $x_1, \dots, x_n \in E$ such that $$\mu = \sum_{i=1}^n \delta_{x_i}$...
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1answer
23 views

Convolution mixture of a probability generating function (population genetics)

I'm trying to work through an old population genetics paper (see here). The following model assumes an infinite number of nucleotide sites and no recombination between different sequences (so you can ...
0
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1answer
43 views

Conditional expectation of a product of random variables

I have two independent continuous random variables $X$ and $Y$ with pdf's : $f(x)$ and $f(y)$ cdf's : $F(x)$ and $F(y)$ a constant $a$ I am trying to express using the given pdf/cdf functions the ...
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1answer
23 views

How to adjust estimation of probability according to new information

Suppose $\{a_1,a_2,\dots,a_n\}$ is a permutation of $\{1,2,\dots,n\}$. The probability of $a_i=j$ is estimated to be $p_{ij}$. The probability matrix might look like this $$ P=\left( \begin{matrix} ...
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0answers
7 views

$X-x_0=O_p(n^{-1/2})$ implies $g(X)-g(x_0)=O_p(n^{-1/2})$

Suppose that $X$ is a random vector and $x_0$ is a fixed vector such that $$ X-x_0=O_p(n^{-1/2}).\tag{$*$} $$ Let $Y=g(X)$ where $g$ has a continuous gradient that is nonzero at $x_0$. Let $y_0=g(x_0)$...
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0answers
45 views

Proving a lower bound on the value of a probability density at the solution to an equation

Assumptions and notation Let $f$ be a twice-differentiable log-concave density function on $[0,1]$, and let $F$ be the corresponding distribution function. Define $x^D$ by: $$\frac{1-2F(x^D)}{f(x^D)}=...
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6answers
580 views

Find the probability of getting two sixes in $5$ throws of a die.

In an experiment, a fair die is rolled until two sixes are obtained in succession. What is the probability that the experiment will end in the fifth trial? My work: The probability of not getting ...
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3answers
82 views

Representing pairwise-independent but not independent occurrences with venn diagram

For $A,B$ and $ C $ partially pairwise independent occurrences (i.e. $I(A;B)=0$, $I(A;C)=0$ ), it is not true to say that $I(A;B,C)=0$, since $I(A;B,C)=I(A;B)+I(A;B|C)$ [<-this is not correct, see ...
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0answers
33 views

Multidimensional change of variables for pdf integration

I have a very simple question, but I could not find the answer, so I have to ask this here: Given is a multidimensional pdf $f(x_1, ..., x_n)$. $x_1, ..., x_n$ are Carthesian coordinates. We want to ...
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1answer
15 views

Find the Expectation of $2\sum\limits_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$ for $y_i$ i.i.d. Cauchy$(\alpha,\beta)$

Find the Expectation of $$2\sum_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$$ given $y_1,y_2...$ iid ~Cauchy$(\alpha,\beta)$ with pdf $(-\infty < y<\infty , \beta>0)$: ...
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1answer
38 views

Problem involving Central Limit Theorem

The following problems is from Durrett 3.4.9: Suppose $X_i$ are independent and $S_n = X_1 + ... + X_n$. Assume that $$ \begin{split} P(X_m = m) &= P(X_m = -m) = m^{-2}/2, \text{ and }\\ P(...
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0answers
35 views

Random walk in a random environment

Consider the random walk in a random environment $\{X_n\}$, that is $P(X_{n+1}=z+1|X_n=z)= \alpha_{z}$ and $P(X_{n+1}=z-1|X_n=z)=1- \alpha_{z}$, where $\{\alpha_{z}\}$ is i.i.d random variables, with $...
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1answer
25 views

Representation of Matrix Calculations - Column Mean Subtract From Each Row

Suppose I have a matrix that looks like this: $$X = \begin{bmatrix}1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0\end{bmatrix}$$ The ...
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1answer
20 views

the relation between the sigma-algebras of two isomorphic spaces [closed]

It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\...
1
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1answer
38 views

$X_i = \mathcal{N}(0, \sigma_i^2)$, with $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$

Let $\{X_k\}$ be independent random variables such that $X_i = \mathcal{N}(0, \sigma_i^2)$, with $\{\sigma_i^2\}$ such that $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$. ...
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0answers
13 views

Rate of convergence for martingales, “merging of opinions” results

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be probability measures on this space. Let $(\mathcal{F}_{n})_{n \in \mathbb{N}}$ be a filtration on $\Omega$. Assuming that the ...
3
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1answer
86 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
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1answer
51 views

What at the chances of getting 20 heads on a row if tossed 100 million times? [duplicate]

I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such ...
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1answer
26 views

Expectation of infimum of asymmetric 1D random walk greater than -$\infty$

I'm reading Durrett's book on Probability and in the example of the asymmetric 1D random walk with $P(X_1=1)=p>1/2$, when trying to compute the expectation of the hitting time $T_{b}:=\inf\{n: S_n=...
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1answer
25 views

Intuition of the Mean wait time in queuing system

In queuing theory, (with a single queue and a single server) , given A is service rate (of customers) and B is arrival rate(of customers) We know that, the average time a customer waits in the ...
3
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2answers
45 views

Calculate the probability in winning tennis game.

A tennis tournament has $2n$ participants, $n$ Swedes and $n$ Norwegians. First, $n$ people are chosen at random from the $2n$ (with no regard to nationality) and then paired randomly with the ...
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1answer
27 views

Conditional probability in independence and mutually exclusive events.

This thread shows that if two events are to be mutually exclusive and independent, one of them should have zero probability. I worked the following example that seems to contradict conditional ...
1
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1answer
36 views

A function is Borel iff $f^{-1}(a,\infty) \in \mathcal{F}$

I'm reading Shao's Mathematical Statistics and part of a proposition is that, if $(\Omega, \mathcal{F})$ is a measurable space then: A function $f$ is Borel iff $f^{-1}(a,\infty)\in \mathcal{F}$ for ...
3
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1answer
85 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
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2answers
38 views

The distribution of a constant $a$ is Dirac mass in $a$, and it's density doesn't exist

Let $X$ a real random variable with density $f_X$ and $Y=g(X)$ with $g$ measurable. I have some trouble understanding the following statement : " If $g(x)=a$ with $a$ a constant, for all $x$, the ...
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1answer
20 views

Find the probability of number plates.

A licence plate consists of a sequence of seven symbols: number, letter, letter,letter, number, number, number, where a letter is any one of $26$ letters ...
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0answers
23 views

Proving a probability concerning a stationary ergodic stochastic process is nonzero.

Let $\{X_t\}_{t\in\mathbb{N}}$ be a stationary ergodic sequence of continuous random variables with full support on the real line. Let $\lambda>1$ and $c>0$. I am interested in the probability $$...
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0answers
27 views

Random walk with drift

Let $X_1,X_2,...$ be independent and identically distributed $\mathbb{Z}-$valued bounded random variables with mean $a=\mathbf{E}[X_1]$, and let $S_n = X_1+\cdots+ X_n$ be the associated random walk. ...
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2answers
51 views

What is the variance of the following random variable? [closed]

There is a box with $5$ rows and $5$ columns, and it contains $25$ pieces. Accidentally, it falls, and the pieces are put back into the box. The number of pieces that will end up in the right place is ...
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0answers
60 views

Basic query related to conditional expectation

I have two variables $X$ and $Y$. I want to find the following probability $$P(X>a(Y+c),X+Y>d)$$ where $a>0,c>0,d>0$. To find the solution I have done following steps $$E_Y[P(X>a(Y+...
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1answer
17 views

Sample space in probability computing

This is a simple example of probability computing. There are $n$ white balls and one black ball in a box. Take the balls one by one out of the box until the black ball appears. Let $X$ denotes the ...
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2answers
31 views

Is my analysis of the following probability problem correct?

I am trying to learn probability, and was reading some lecture notes, while I came across the following example- So I tried to understand the problem and the solution, For sampling without ...
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0answers
13 views

Expectation of $Y=X\mathbf 1_{X>t}$ in terms of the CCDF $P(Y>x)$

I have random variable $Y=XU(X-t)$ (here $U(X)$ is the unit step function, $Y$ has non-negative support and depends on other random variable $X$) which has a CCDF $P(Y>t)$. I want to write the ...
3
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2answers
253 views

Expected value of the zeros of random polynomials of degree two

Let $a_1,a_0$ be i.i.d. real random variables with uniform distribution in $[-1,1]$. I'm interested in the random zeros of the polynomial $$p(x) = x^2 + a_1x + a_0. $$ One thing (between many) thing ...
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4answers
40 views

Can anyone explain the convolution of independent random variables?

Specifically, suppose that $X_1$ and $X_2$ are independent and $f$ is a Borel-measurable function s.t. $f(x_1, x_2)=\begin{cases} 1, & \text{if } x_1+x_2 \leq x\\ 0, & \text{otherwise} \end{...
6
votes
1answer
49 views

Example of set, finite outer measure, subsets, where outer measure does not converge

What is an example of a set $X$ and a finite outer measure $\mu^*$ on $X$, subsets $A_n \uparrow A$ of $X$, and subsets $B_n \downarrow B$ of $X$ such that $\mu^*(A_n)$ does not converge to $\mu^*(A)$ ...
0
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1answer
43 views

Can we use a symmetry argument instead of integration in BASIC probability?

Suppose $H$ is a random variable with pdf $f_H(h)$. Let $X$ and $Y$ be random variables with joint pdf $$f_{X,Y} = f_H(x) f_H(y)$$ Prove $$P(X \ge Y) = 1/2$$ Is it possible to ...
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0answers
20 views

Finding best strategies in a problem about traffic lights

A problem came up in a course of a conversation with my friend. Suppose we have a street with several traffic lights, placed equidistantly one from another. Time of them being green $t_g$ and time of ...
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0answers
38 views

Prove that if $X$ and $Y$ are independent, then $h(X)$ and $g(Y)$ are independent in BASIC probability — can we use double integration?

In advanced probability we can just say: \begin{align} & P(h(X) \in A, g(Y) \in B) \\[6pt] = {} & P(X \in h^{-1}(A), Y \in g^{-1}(B)) \\[6pt] = {} & P(X \in h^{-1}(A)) P(Y \in g^{-1}(B)) \...
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1answer
57 views

Probability of $|H-T|$ in 10,000 coin tosses

If a fair coin is thrown $10,000$ times. Using the binomial convergence to normal,find $P|H-T|\le 80$ My intuition say that mean is 0.But I am not able to proceed further.
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1answer
21 views

Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...
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votes
2answers
43 views

$\mathbb{E}[|X|^n] < +\infty \implies \mathbb{E}[|X|^k] < +\infty, k \leq n$

Show that if $\mathbb{E}[|X|^n] < +\infty$, then $\mathbb{E}[|X|^k] < +\infty, \forall k \leq n$. I guess I have to apply Hölder Inequality, but I was not able to find out what $p$ and $q$ are ...
0
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1answer
19 views

Exponential martingale, Lévy-process and stopping times, definition quesiton.

I feel there is some ambiguity for the definition of the exponential martingale for a levy process which I do not understand. For a Lévy process it can be shown that $E[e^{iuX_t}]=e^{t\eta(u)}$, ...
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1answer
31 views

Transformation of density and $W=(X+Y+Z)^2$

I want to solve this exercise with the transformation formula, what did I do wrong in my solution?: Let $X,Y,Z$ be independent random variables with uniform distribution on [0,1]. What's the ...
0
votes
1answer
26 views

Interesection of ranges r.v. and max of r.v.

I have the following question, let $X_1,...,X_n$ be some discrete random variables, and let $Y$ and $Y'$ be two geometric random variables that are not necessarily independent of $X_i$. let $p_Y \geq ...
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0answers
14 views

Partition Probability Proof

I'm tasked with proving the following Lemma: Let X be a set of $n^1$ < n elements, and B $\subset$ X, |B| = k. Suppose $P_1, P_2,...P_r$ are random partitions of X, where each $P_i$ partitions X ...
2
votes
0answers
44 views

Almost sure convergence (correctness of an argument)

Is this statement correct? If $X_n \xrightarrow{a.s} c$, where $X_n$ is a sequence of random variables and $c$ is a constant, then we can conclude that since almost sure convergence implies on ...
1
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1answer
63 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
0
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0answers
31 views

Exercise from David Williams' “Probability with Martingales” page 25

I was stuck at the exercise page 25 and then I found the answer here : Noob Question : Need help to understand : Probability with Martingales : page 25 But there is still one point I don't get, how ...