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

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0
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2answers
20 views

convergence in distribution of truncated gaussian variables

Let $X$ be a random variable which is distributed normally with mean $\mu=0$ and variance $\sigma=1$. Suppose that $X_n$ is a random variable for any positive integer $n$ with truncated normal ...
0
votes
2answers
20 views

Effect on probability of adding a constant to the random variable

I have this question in my notebook.A Drunk person performs a random walk over positions $0,\pm1,\pm2,\dots$ He starts at 0, he takes successive 1 unit steps going to the right with probability p and ...
0
votes
1answer
67 views

Is the expected value of the difference of these two random variables, with infinite expected value, $0$, or undefined?

Let's say we have two independent random variables, $x_1$ and $x_2$, both have a probability mass function $X$ defined as $$X(n) = \begin{cases} 2^{-m} & \text{if $n=2^m$ for $1 \le m \in \mathbb ...
3
votes
1answer
25 views

Finding a random variable $X$ such that $X_n$ (given) converges in distribution to $X$

For every $n\in\Bbb{N}$, let $X_n$ be a random variable which gets the values $\{-1, -\frac{n-1}n,...,-\frac 1 n, 0, \frac 1 n,...,\frac{n-1}n, 1\}$ with equal probability. Find a random ...
0
votes
2answers
35 views

CDF from PDF when $P(X \ \text{is} \ \text{even}\mid X\geq4)$

Given a PMF p by x 2 3 4 5 6 p(x) 0.1 0.2 0.2 0.3 0.2 And let X be a random variables with values in the set {2,3,4,5,6} Is it correct to ...
1
vote
0answers
30 views

Random Samples and Sample Variance Bound

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population. Show that: $$\max_{1 \leq i \leq n}|X_{i}-\bar{X}|<\frac{(n-1)}{\sqrt{n}}S$$ Where we have the sample variance ...
0
votes
0answers
20 views

Geometry of Vector Random Variables and Joint Distribution

I'm not a statistician but have been trying to understand the following problem in my research: I have two $3\times 1$ random vectors $\mathbf{v}$ and $\mathbf{w}$, and a function ...
0
votes
1answer
19 views

Calculate $P(X\leq 1\mid Y\leq0)$

I need to calculate $P(X\leq1\mid Y\leq0)$ I've found that $$P(X\leq1\mid Y\leq0)=\frac{P(X\leq1,Y\leq0)}{P(Y\leq0)}$$ But is it true that ...
0
votes
2answers
34 views

Doubt in Conditional Probability

I'm studying Information theory from the book Information Theory, Coding and Cryptography-Rajan Bose. I got confused at one pos where they have derived the equation ...
0
votes
0answers
20 views

Implications of symmetric probability density function

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with probability density function $f$. Suppose $f$ is symmetric around zero. This ...
0
votes
0answers
19 views

Bernoulli Random Variables with Pairwise Negative Correlation

I was wondering if there is a simple way to generate Bernoulli Random Variables that have negative correlations pairwise with a lower bound on the success probability? If that isn't possible, then ...
1
vote
0answers
27 views

How can I compute the number of selected green ball from a given selection prob.

I have $2$ red balls in box 1 and $4$ green balls in box 2 as figure. The prob. selection the red balls (R) in box 1 is $$P(R=1)=0.1$$ $$P(R=2)=0.9$$ And prob. selection the green balls (G) in box ...
1
vote
0answers
24 views

Expected length of a path in a hyper graphs

Assume in a graph we represent each node by k bits. Rout is a path that connect a node with its destination and is unique. for example, from node 000 to node 111 we need a rout with length 3 and that ...
1
vote
0answers
41 views

Convergence of a sum of random variables with Bernoulli coefficient.

I present a problem which is connected with some of my previous questions. Suppose that $Y_t$ is a "regular enogh" (for example $Y_t=W_t$ with $W_t$ a Brownian motion) stochastic process with ...
1
vote
0answers
18 views

calculation of the maximum likelihood estimator

I am given n random variables $ y_1 = \theta_1 \theta_2 + e_1$ $ y_i = \theta_1 + e_i$ for $ \space i = 2,...,n$ where $\theta = [ \theta_1 \space \theta_2] ...
1
vote
1answer
30 views

Limit of Snedecor's F

Suppose we have a random variable $X$ such that $X\sim \dfrac{d}{n-d}F(d,n-d)$, with $d,n\in\mathbb{Z}$. What happens when $n\to\infty$? And when $d\to\infty$? I think when $n\to\infty$ then it goes ...
1
vote
0answers
16 views

Convergence of a sequence of Bernoulli variables.

For $\lambda\in(0,1)$ consider the following sequence of Bernoulli random variables $$ \mathbb{P}\left[B_n=1\right]=1-\frac{\lambda}{n},\quad \mathbb{P}\left[B_n=0\right]=\frac{\lambda}{n}. $$ Now ...
0
votes
0answers
33 views

Gcd and i.i.d. random variables

As I was working for my test on random variables I ran into the following problem that I can't solve : Let $X$ and $Y$ i.i.d. following the discrete uniform distribution for n a natural integer. ...
0
votes
1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
1
vote
1answer
15 views

Transformation related to a particular random variable

Consider a random variable $Y$ with the density function given by $$f_Y(y) = \left\{\begin{array}{cc} 2y^3 & -1 \leq y \leq 1 \\ 0&\text{otherwise.} ...
1
vote
1answer
28 views

Find the maximum and minimum value of variance. [duplicate]

Let $X$ be an arbitrary random variable takes values in $\{0,1,2,...,10\}$. Then the minimum and maximum values of variance of random variable $X$ are $0$ and $30$ $1$ and $30$ $0$ and $25$ $1$ ...
0
votes
1answer
17 views

Probability of terms involving dependent random variables

I want to calculate the probability as \begin{equation} \mathcal{A} = \mathbb{P}\left(B X > \zeta_s, CXY > \zeta_s\right), \end{equation} where B, C, and $\zeta_s$ are positive constants, while ...
0
votes
1answer
45 views

Existence proof for a random variable $X$ where $\operatorname{E}[X]$ exists, and $\operatorname{E}[X^2]$ doesn't

Can you think of a random variable $X$ where: $\operatorname{E}[X]$ exists, and $\operatorname{E}[X^2]$ doesn't? I'm not sure if I remember correctly but I remember having heard in the lecture that ...
1
vote
1answer
51 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
votes
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
votes
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
vote
1answer
41 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
votes
1answer
45 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
votes
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 ...
0
votes
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
vote
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
vote
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 $s$ ...
-1
votes
2answers
40 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
votes
1answer
30 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
220 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
votes
1answer
29 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 ...
-1
votes
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 ...
0
votes
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
91 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
vote
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
votes
1answer
40 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 ...
-1
votes
1answer
42 views

gaussian density and 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 ...
0
votes
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 ...
1
vote
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
votes
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 $$P(X=1)=0.1$$ $$P(X=2)=0.5$$ $$P(X=3)=0.2$$ ...
1
vote
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 ...
0
votes
0answers
16 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
votes
1answer
63 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 ...
0
votes
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 ...