Questions about maps from a probability space to a measure space which are measurable.

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1answer
52 views

Conditional expectation in Poisson point process

Considering a Poisson process with parameter $\lambda$, let $N(t_2)$ denote the number of events in $(0,t_2]$ and $N(t_1, t_3)$ denote the number of events in $(t_1,t_3]$, under the assumption that ...
0
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1answer
55 views

Does $P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3\right)$?

This link shows that that $P(X_i = x) = 0$ so can we say, \begin{equation} P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3\right) \end{equation} Assumptions $X_i, X_j$ are random ...
2
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0answers
26 views

Convergence of infinite-dimensional random variables

For every $n \in \mathbb{N}$ and every measurable $E \subseteq [0,1]$, the object $f_n(E)$ is a random variable that takes real values. The sequence ($f_n$) can thus be understood as a sequence of ...
1
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1answer
42 views

Random Variables, Minimize Variance

The variance of $X_1$, $X_2$ are 1 and 4, and the correlation coefficient p=-0.3 1)Calculate the variance of $Z_1 = 2X_1+X_2$ 2)Calculate the value of a that minimizes the variance of $Z_2 = ...
0
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1answer
46 views

Variance of squared random variable

Can anyone help to prove this equation for any distribution $$ E(z^4)=1+\operatorname{Var}(z^2) $$ where $z$ is a random variable with the standard normal distribution $$z=\frac{x−μ}σ$$
2
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0answers
30 views

Finding the probability of a region $|X-Y|$

I have a region in a 3-D space with a density of $$ \ f_{x,y,z}(X,Y,Z) = \begin{cases} 1 & \text{if $ (x,y,z)\in W$}; \\ 0 & \text{if $(x,y,z)\notin W$};\\ \end{cases} \ $$ Being $W$ the set ...
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1answer
23 views

Covariance matrix of a random vector

According to the documentation of Matlab regarding the function cov(): "if A is a vector of observations, then cov(A) = C is the ...
1
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1answer
15 views

Finding the probability of a region inside a pyramid

I have a region in a 3-D space with a density of $$ \ f_{x,y,z}(X,Y,Z) = \begin{cases} 1 & \text{if $ (x,y,z)\in W$}; \\ 0 & \text{if $(x,y,z)\notin W$};\\ \end{cases} \ $$ Being $W$ the set ...
1
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1answer
27 views

Prove that there exists $s > 0$ such that $E[X^s] < 1$ given $E[X^r] < ∞$ for some $r$ and $E[\log X] < 0$

Let $X$ be a random variable with $X ≥ 0$ a.s. and such that $E[X^r] < ∞$ for some $r > 0$ and $E[\log X] < 0$. Prove that there exists $s > 0$ such that $E[X^s] < 1$. I know when $s$ ...
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2answers
41 views

Finding out the constant in p.d.f. with given mean?

Probability density function $$f(x)=\alpha\ e^{-x^2-\beta\ x},\ -\infty<x<\infty$$ Also $E(X)=-\frac{1}{2}$ I tried solving it using respective formulas of total probability equal 1 and the ...
0
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1answer
32 views

Proof of Hoeffding's Covariance Identity

Let $X,Y$ be random variables such that $\text{Cov}\left(X,Y\right)$ is well defined, let $F\left(x,y\right)$ be the joint-CDF of $X,Y$ and let $F_{X}\left(x\right),F_{Y}\left(y\right)$ be ...
2
votes
3answers
226 views

Confusion about the range of the sum of i.i.d. random variables

Let $X_1, X_2, ...X_n$ be independent and uniformly distributed random variables on the interval $[0,1]$. Now suppose I wanted to calculate the probability density function of $Z = X_1 + X_2 + ... + ...
0
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1answer
30 views

Integral of a distribution function

I am attempting to prove the following identity for the random variable R defined on ($ -\infty $, $ +\infty $). Upon attempting to integrate by parts I run into an indeterminate form. I am not sure ...
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1answer
18 views

What is the correlation between the pairwise differences of 2 bivariate normal random variables? [closed]

Given (X,Y) bivariate normal, $U = \frac{X_i - X_j}{\sqrt2\sigma_x}$ and similarly $V = \frac{Y_i - Y_j}{\sqrt2\sigma_y}$ for any two independent pairs $(X_i, Y_i)$ and $(X_j, Y_j)$. Why is this true ...
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0answers
13 views

Find probability distribution to find minimum distance between four RV.

Consider, we have four random variables $X, Y, -X $ and $-Y$, where $X$ and $Y$ are circularly symmetric complex normal random variables. Now let four distances $D_1=dist(X,-X), D_2=dist(X,Y), ...
2
votes
3answers
92 views

Moments of minimum of random variables

Let $\mu$ be a non-atomic probability measure on $[0,\infty)$ and sample $X_1,X_2$ from $\mu$ independently. Does $\min(X_1,X_2)$ have twice as many moments as $X_1$? Is the quantity $$ \frac{\mathbb ...
1
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1answer
22 views

Proving $P[a\lt X \lt b]=F_X(b)-P[X=b]-F_X(a) $.

X is a random variable then :$$ \{X\lt b\}= \{X\lt a\} \cup\{a<X<b\}$$ and $$\{X\lt a\} \cap\{a<X<b\}=\phi. $$ Hence $F_X(b)=P[a\lt X\lt b]+F_X(a) $. Can anyone tell where I am doing ...
0
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1answer
41 views

The prob. distribution of sum of two independent random variables

I have two Random variables $X,Y$, They are independent. In which, $X,Y$ follows same distribution $$P(X=1)=P(Y=1)=0.1$$ $$P(X=2)=P(Y=2)=0.4$$ $$P(X=4)=P(Y=4)=0.3$$ $$P(X=10)=P(Y=10)=0.2$$ How ...
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2answers
45 views

Empirical Process estimation using gaussian density and specific random generator

EDITED: To formulate into math framework: I have a sampling generator producing IID gaussian. To highlight the convergence in the distribution, I calculate the following error. Given a precision ...
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0answers
21 views

Domain and codomain of function of random variables

I have a question related to the exact definition of a function of random variables. In some sources I have found that "a function of a random variable has as domain the sample space induced by the ...
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0answers
27 views

Probability space induced by a random variable

Consider three random variables $Y$, $X$, $Z$ all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $Y: \Omega \rightarrow \mathcal{Y} \subseteq \mathbb{R}$, $X: ...
0
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1answer
27 views

Calculating characteristic function of random variable

I would like to calculate the characteristic function of $Z_{\beta, n}=(1-\beta^2)^{1/2}\sum_{k=0}^n\beta^kX_k$, where $X_i$ are independent random variables with $P(X_i = 1)=P(X_i=-1) = 1/2$ and ...
2
votes
2answers
52 views

Find prob. that only select red balls from $n$ (red+blue) balls

There are 4 blue balls and 6 red balls(total 10 balls). $X$ is a random variable of the number of selected balls(without replacement), in which $$P(X=1)=0.1$$ $$P(X=2)=0.5$$ $$P(X=3)=0.2$$ ...
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0answers
19 views

Uniform distribution density function

Let's say we have two random variables $T_1$ and $T_2$ and the joint density function of the two is uniform over the region $0\leq t_1\leq t_2 \leq L$, where L is a positive constant. Then the area of ...
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0answers
17 views

Definition of random variable on the Euclidean Space

Consider a random variable $X: \Omega \rightarrow A \subset\mathbb{R}^k$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and inducing the probability space $(\mathbb{R}^k, ...
2
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1answer
64 views

Kolmogorov's 0-1 law - Corollary from Loève's book

Studying from the book Probability Theory by Michael Loève I came across the following corollary of Kolmogorov's 0-1 law, which is not proved: Corollary. If $X_n$ are independent r.v.'s, then the ...
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1answer
32 views

Money problem…probability spend in particular time

A child puts money in piggy bank every day , in particular 10 , 20 , 30 , 40 , 50 , or 60 cents with the same probability . Find the probability of spending at least 80 days before having collected 30 ...
0
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1answer
20 views

finding the probability of not being answered more than two calls!

In a call center , the time between successive calls is exponential random variable with $\mathbf{E}X$= 2 minutes . If the operator is removed for 5 min , what is the probability of not being answered ...
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0answers
15 views

Distribution of the norm of uniform random unit vector after linear transformation

Suppose that $\mathbf{u}$ is a uniform unit vector. It is obtained as $\mathbf{u}=\frac{\mathbf{n}}{||\mathbf{n}||}$ where $\mathbf{n}$ is a white Gaussian vector. Clearly we have ...
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2answers
52 views

Simple Monte-carlo approximation of $\pi$ and integration using Matlab.

Hi I am new to programming and the following task that my tutor want me to handle was a bit to hard to begin with. I hope someone can help me and I need the code in Matlab. You are throwing darts ...
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0answers
46 views

Can you explain Gärtner-Ellis Theorem?

At lecture it's just been stated and that's it, but I want to understand the idea behind the theorem. EDIT (Well I deserved the downvote, I guess. Let's make the question clear) My version of ...
1
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1answer
18 views

Reference on properties of binary random vectors [closed]

I am studying random vectors of length $n$ for which each element is an i.i.d Bernolli random variable with probability $\frac{1}{2}$. I think that many useful and interesting theorems can be proved ...
5
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2answers
114 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
1
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1answer
65 views

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}$dx?

Characteristic function of $X^2$ where $X: \mathcal N(0,1)$. $$\int_{-\infty}^{+ \infty} e^{itx^2}\frac{1}{2 \pi}e^{-\frac{x^2}{2}}dx?$$ I just need to solve this integral. But, I don't know how. ...
0
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1answer
13 views

Mean square convergence not higher-order mean

There are numerous examples of sequences random variables $\{X_n\}_n$ converging in mean to a random variable $X$, i.e., $\mathbb{E}[|X_n-X|]\to 0$, but not in mean square, i.e., ...
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0answers
17 views

Probability of Modulus of a sum of two symmetric random variables

Suppose that $X$ and $Y$ are independent, symmetric and identically distributed random variables. Let $$P\left(\left|X+Y\right|\le K\right)>A$$ for $K, A$ constants. Show that there exist ...
1
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1answer
22 views

Using method of maximum likelihood find the estimator for $\mathcal N(m,1),m<0$ and $\mathcal U(\theta, 1), \theta<0$

Using method of maximum likelihood find the estimator for $\mathcal N(m,1)$-normal distribution and $\mathcal U(\theta, 1), \theta<0$ From what I understand, if the parameter is negative it is ...
0
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0answers
22 views

Pythagoras' Theorem for Random Variables?

Let $X$ be a square-integrable random variable on $(\Omega, \mathcal{A}, P)$. Let $\mathcal{F}$ be a sub-sigma-Algebra and let $Z$ denote $Z:=E[X \mid \mathcal{F}]$. We know, since the conditional ...
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1answer
39 views

equivalence of properties of characteristic function of a random variable

I would like to prove that for a random variable $X$ and its characteristic function $\phi_X$ the following three properties are equivalent. $i) \ \phi_X(s) = 1$ for some $s \neq 0$ $ii) \ \phi_X$ ...
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1answer
29 views

Word problem about adding continuous random variables

I cannot seem to figure out how to do the following word problem: "The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 oz and a SD of 0.3 oz. A large bowl holds a mean ...
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2answers
21 views

Is the joint distribution of these two dependent Gaussian RVs, Gaussian?

I have two dependent Gaussian variables $X_1,X_2$ with unit mean each, and standard deviations $\sigma_1=2a$ and $\sigma_2=a$ respectively, while $a>0$ Is the joint distribution of these two ...
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2answers
21 views

Sample mean - two definitions

There are two definitions of sample mean. First one Given $n$ independent random variables $X_1$, $X_2$, ..., $X_n$, each corresponding to randomly selected observation. The sample mean is defined as ...
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1answer
27 views

Math Behind Producing Uniform Distribution

I am familiar that if one can produce a uniform distribution, doing so, one can then produce random numbers for other types of distributions. I have tried reading some articles online but I am still ...
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0answers
32 views

How to derive Erlang distribution from the Exponential distribution?

I read that for a random variable $X$ to have Erlang distribution, it will be the sum of identical random variables with exponential distribution, but i cant derive the formula. The density of an ...
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1answer
31 views

Conditions implying uniform integrability

We say that a family of random variables $X_n, n \geq 1$ is uniformly integrable if $$\lim_{M \rightarrow \infty} \sup_{n} E[|X_n| 1_{|X_n|>m}]=0.$$ I am struggling with some proofs and could ...
2
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1answer
20 views

Why does this “wlog” make sense: $L^p$-norms of random variables

Let $$\overline{X_n}:=\max_{0 \leq m \leq n} X_m^+$$ for a sequence of random variables $X_i, i \geq 1$ (in fact, it is a submartingale), where $X_m^+:=\max(X_m,0)$. We want to show that ...
0
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1answer
24 views

When the union probability of two random variables becomes equal to one? Is it only when one of them is certain?

Let's say I have the two events $X_1 < T_1$ and $X_2 >T_2$, Where $X_1,X_2$ are two dependent Gaussian variables, and $T_1,T_2>0$. I want to know when the union probability of the two events ...
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1answer
40 views

Matching problem expectation and variance

The matching problem: Suppose $n$ gentlemen go out for dinner and leave their hats in the cloakroom. After the dinner (and several glasses of wine) they pick their hats completely randomly. Denote ...
7
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1answer
89 views

A.s. equality between limsup of random variables

"Let $(X_n)_{n\ge 1}$ be a sequence of uniformly bounded random variables defined on a probability space $(\Omega, \mathscr{F}, P)$. Moreover define $\mathscr{F_0}=\{\emptyset,\Omega\}$ and ...
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0answers
24 views

On the definition of probability function of discrete r.v. in DeGroot & Schervish “Probability and Statistics” (relation to probability mass function)

I am reading DeGroot & Schervish’s text “Probability and Statistics”, and when it gets to defining the probability function of a discrete random variable $X$, they write: The probability ...