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

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1answer
34 views

Show: $X_n\xrightarrow{\mathcal{d}} X$, then $\mathbb{E}\lvert X\rvert\leqslant\liminf_{n\to\infty}\mathbb{E}\lvert X_n\rvert$

Let $X_n, X$ be random variables with $X_n\xrightarrow{d} X$. Show that then $$ \mathbb{E}\lvert X\rvert\leqslant\liminf_n \mathbb{E}\lvert X_n\rvert. $$ So let $X_n\xrightarrow{d} ...
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2answers
65 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
1
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1answer
31 views

Geometric random variable $X$, $Pr(X\ is\ even) =$?

Original Question: Toss an unfair coin until we get HEAD. Suppose the total number of tosses is a random variable $X$, and $Pr(HEAD) = p$. What is the probability that $X$ is even? Denote this event ...
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1answer
34 views

variance of a random variable

If $X_1, X_2 , ....., X_n$ iid $N(0,1)$ , and $S^2$ was defined as the population standard deviation we are to find the variance of $S^2$ I want to know the distribution in order to find the ...
0
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2answers
55 views

moment generating function technique

If $X$ was a random variable with a distribution $\mathrm{Normal} ( 0, 1 )$, using moment generating function technique we have to show that $Y= X^2$ has the Chi-square distribution with $1$ degree of ...
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2answers
22 views

existence of a RV with distribution given by a linear combination of other distributions

Question: Let $X$ and $Y$ be random variables defined on a $(\Omega,\mathfrak{F},\mathbb{P})$ probability space with distribution functions $F_X(t)$ and $F_Y(t)$, respectively. (a) Show that for any ...
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2answers
49 views

Sampling from a Normal Distribution

If I am sampling randomly from only the -sigma to +sigma interval of a normal distribution and rejecting all other numbers, does it imply that the probability density changes? If so, by what degree? ...
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0answers
33 views

Random variables plus a constant

Quick question: If a random variable $\mathcal{Z}$ with expected value 0 and $\sigma=10$ Volts, how do you take into account a constant voltage of 5 Volts added to it? Do I just add 5 to the density ...
0
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2answers
20 views

Variance of three possible outcomes

I am new to this kind of things so maybe you could help me get the reasoning. I have a continuum of outcomes on the interval $[0,1]$. Now, let us cut the interval into two pieces so that there are two ...
0
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0answers
42 views

Absolute value of the Fourier Transform of Gaussian random variable

Assume you have a normally distributed random variable $x$ with zero mean $\mu$ and standard deviation $\sigma$. Now you take the Fourier transform of it. The resulting complex random variable ...
0
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1answer
33 views

$(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots) $ are independent $\Rightarrow (X_t)$ are independent

In some lecture notes in Time Series Analysis it was written: If the random vectors $(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots)$ are independent for all $t \in \mathbb{Z}$, then it is ...
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0answers
22 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
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2answers
43 views

Can any sequence of RVs converge almost surely to a non-constant RV?

We know for almost sure convergence, $X_n \rightarrow X \text{ a.s. as } n\rightarrow \infty$ if $$ P\{\limsup_{n\rightarrow\infty}X_n = \liminf_{n\rightarrow\infty}X_n\} = 1. $$ Then does this mean ...
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0answers
13 views

The norm of a vector with gaussian noise

Say I have a vector of length n, $v \in R^n$ where $0<=v(i)<=1$ for each i, Now, I add noise: let $n$~$Normal(0,\sigma)$ And to each i I add noise v(i)+n , such that the noises are independent ...
2
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1answer
44 views

Calculating joint MGF

This is an end-of-chapter question from a Korean textbook, and unfortunately it only has solutions to the even-numbered q's, so I'm seeking for some hints or tips to work out this particular joint ...
10
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1answer
125 views

Why is the function $\Omega\rightarrow\mathbb{R}$ called a random variable?

I do not understand the relation of a normal variable "x", which is to me just a placeholder for an element of a set, and a random variable, which is a mapping from the set of all possible events to ...
0
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1answer
25 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
2
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0answers
81 views

Coupling Pairs of Random Variable.

Let $\{X_i\}_{i=1}^{n}$ and $\{Z_i\}_{i=1}^{n}$ be sets of independent random variables with coupling $\{X^{\hat{}}_i\}_{i=1}^{n}$, $\{Z^{\hat{}}_i\}_{i=1}^{n}$ respectively. It then states ...
1
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1answer
33 views

Random variable transformation function

I am stuck with a random variable transformation problem ($Y=\phi(X)$). The random variable $X$ has a uniform distribution $U(-1,1)$, and I want to transform it into $Y$ which is also an uniform ...
0
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1answer
34 views

Prove that ${(1-p+pe^t)}^n = E(e^{t\sum Z_i}))$

Let $Z_1,...,Z_n$ be independent random variables such that $ Z_i \in \vert0,1\vert$. Prove that ${(1-p+pe^t)}^n = E(e^{t\sum Z_i}))$ where $p=\sum{\frac{E(Z_i)}{n}}$ Not quite sure how to do this. ...
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0answers
15 views

Quantile function for multivariate random variable

Can anyone pls help me in resolving this issue? Let $w_k \sim \mathcal{N}(0,\mathbb{I}_{n\times n} )$ be a Gaussian mulativariate random variable, $p\in[0,1]$, $G\in \mathbb{R}_{r\times n}$, $H\in ...
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0answers
46 views

If $X$ is a random variable, under which conditions is $g(X)$ also a r.v.?

In many instances, functions of random variables appear, and we usually treat them as random variables also. In the 3d edition, pp. 85-86, of this well-known book (now in its 4th edition), we find the ...
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1answer
21 views

Mean of a PMF with a variable

I am given the following PMF: and I am asked to find the mean. I'm a bit lost on what to do with the ...
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0answers
48 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
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2answers
63 views

Variance of the sum of Bernoulli Random variables?

$\newcommand{\var}{\operatorname{var}}$ Let $X_{i}$ be a Bernoulli random variable with paramater $p_{i}$ where $p_i$ itself is a random variable that ranges from $0$ to $1$. The expectation of ...
2
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1answer
31 views

A basic doubt on sigma algebra generated by a random variable

Why do we need the concept of sigma algebra generated by a random variable ?
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2answers
189 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
1
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1answer
38 views

How to show that a sequence of random variables doesn't converge in probability?

Say, we have the sequence of random variables defined on $\Omega=[0,1]$ with uniform distribution: $$X_n(\omega) := \begin{cases} \omega, & \text{if $n$ is odd} \\ 1-\omega, & \text{if $n$ ...
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0answers
13 views

Robbins-Siegmund like theorem for a nonlinear system

Recently I came to know about this theorem due to Robbins and Siegmund which states the following: Let us have on a probability space $(\Omega, \mathcal{F},P)$, a filtration $\{\mathcal{F}_n\}$ and ...
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1answer
39 views

Show that the following procedure generates a random variable $X \sim p_{X|Y}(x|y)$

Suppose that $X$ and $Y$ are discrete random variables with a joint probability mass function $p_{XY}(x, y)$. Generate $X \sim p_X(x)$ Accept $X$ with probability $p(y|X)$ If $X$ is accepted, ...
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0answers
25 views

Conditional Variance of a Random Variable?

Say I have the following random variable: Where $\varphi_{i}$ and $\rho_{j}$ are themselves independent random variables (note: given particular values for $\varphi_{i}$ and $\rho_{j}$, $Q_{ij}$ is ...
0
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1answer
46 views

Question about Borel sigma algebra

I have a question regarding singletons (unit set) in a Borel-sigma algebra: Lets say that K is a σ-algebra on a random range [0,100]: A. Is it true that i can take any singleton in this range ...
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0answers
9 views

Estimating variance from the sequence

Suppose that we have $\{X_n\}\to X\sim N(0,\Omega)$ where $X_n$ can be obtained from observations. My problem is to estimate $\Omega$ consistently. If $var X_n$ converges to a "finite" matrix, then ...
0
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1answer
27 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
3
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1answer
83 views

Proving increasing function defined as bivariate normal

Suppose $c>0,\sigma>0$ and $\tau>0$ are fixed real constants. Then I'd like to prove that the function $g_c:(-1,1)\mapsto\mathbb{R}$ defined by \begin{equation} ...
1
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1answer
16 views

Preservation of positivity under convergence in distribution?

I have the following situation: $\mathbb{P}(X_n\geq 0)=1\quad \forall n\in \mathbb{N}$ and $X_n \overset{\mathcal{D}}{\rightarrow} X$ as $ n\rightarrow \infty$. How do I prove that the positivity ...
0
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2answers
178 views

Are any linear combination of normal random variables, normally distributed?

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed. It is also said that if (x1,x2,..,xn) is ...
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2answers
19 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
0
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1answer
36 views

chi-square random variable pdf

Hi could anyone please help me in computing the bellow integral $$\int_{0}^{\tau}\frac{y^{-1/2}e^{-y/2}}{2^{1/2}\gamma(1/2)}dy$$
0
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1answer
19 views

Generate random number from set

You are given a set T of n non negative real numbers ${t_0, t_1 ... t_n}$ and probabilities $p_0,p_1...p_n$ where $\sum\limits_{i}^{n} p_1 = 1$. Assume $t_0 < t_1 ...< t_n$. Given a random ...
0
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0answers
64 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
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4answers
111 views

Why does maximum likelihood estimation for uniform distribution give maximum of data?

I am looking at parameters estimation for the uniform distribution in the context of MLEs. Now, I know the likelihood function of the Uniform distribution $U(0,\theta)$ which is $1/\theta^n$ cannot ...
1
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1answer
56 views

Distribution of the first passage time of a Gaussian random walk

Does anyone know the distribution for the first passage time of a Gaussian random walk i.e. $$ S_n = \sum_{i=1}^n X_i $$ where $X_i$ are iid normally distributed random variables. The first passage ...
0
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2answers
47 views

the maximum of two random variable

The maximum of two random varibles $X$ and $Y$ is: $$Z=\max\{X,Y\}= \begin{cases} X & \text{if } X \geq Y \\ Y & \text{if } Y \geq X \end{cases}$$ I don't understand. So if I roll two dice, ...
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2answers
70 views

Eigenvalues of a Random Matrix

I am studying the theory of random matrices lately, but there is a basic issue troubling my life. I hope someone here explain me this, thank you. A random matrix is defined as a matrix whose entries ...
2
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1answer
39 views

Is the product of two independent uniform integrable random variable is uniform integrable?

Is the product of two independent uniform integrable random variable is uniform integrable? What is the role independence plays here?
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28 views

Expectation of product of random vectors

Let $\mathbf{x} \sim \mathcal{N}(\mathbf{\mu},\mathbf{\Sigma})$ be a random vector, that is normally distributed. How can one calculate the following expression? $$ \mathbb{E}[\mathbf{xx}^T]$$ And ...
0
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0answers
18 views

Efficient algorithm for point estimation of a dependent random variable

Suppose $X$ is a normal-distributed random variable and $f$ is a known smooth function (possibly quite complicated, with many oscillations). Let $p(y)$ be the pdf of the dependent random variable $Y = ...
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1answer
78 views

help with Borel Cantelli lemma

There is a sequence of random variables $X_1,X_2,...$ For each i $X_i$ ~ $Normal(0,1)$ Is $ \frac{X_n}{n} \rightarrow 0 $ almost surely? Is $ \frac{X_n}{lnn} \rightarrow 0 $ almost ...
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0answers
11 views

Any literature available on Functions of Discrete Random Variable?

There is a good literature available on Functions of Continuous Random Variables, but is it true for the discrete case? Let discrete random-variable $Y=cX$ , where $c$ is positive integer and $X\sim ...