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

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4
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1answer
46 views

Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r| $$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r $$ Attempt I ...
1
vote
1answer
47 views

Random variables with joint density function

Let R be the rectangle $\ \{(x, y); 0 <= x <= 2, 0 <= y<= 1\} $, and let $\ f(x, y) = >k(x^2+ y^2)$ on R and zero elsewhere. (a) Find the value of k which makes f a joint ...
0
votes
1answer
36 views

Does higher variance imply a higher covariance?

Suppose I have three random variables A,B,C. if var(B) > var(C) does that mean cov(A,B) > cov(A,C)? Assuming neither is uncorrelated meaning cov(A,B) and cov(A,C) don't equal 0.
1
vote
1answer
23 views

A mapping that does not preserve convergence in distribution

I'm trying to come up with a map $H: \mathbb{R}^k\to \mathbb{R}^k$ and a sequence of random vectors $X_n\Rightarrow X$ in $\mathbb{R}^k$ for which $H(X_n)$ does not converge in distribution to $H(X)$. ...
0
votes
0answers
16 views

How can I visualize the sample space of a complicated random experiment?

A sample space $\Omega$ of a random experiment is the set of all possible outcomes $\omega$. When dealing with random variables (which, formally, is a map from $\Omega \mapsto R$), we usually operate ...
0
votes
1answer
33 views

How to deal with the following problem of correlated random variables?

I have the following information: $\left[ \begin{array}{l} {X_1}\\ \vdots \\ {X_K} \end{array} \right]$ are correlated random variables with (zero mean, unit variance) covariance matrix $\left( ...
1
vote
2answers
63 views

Maximum likelihood estimator?

I am looking at some questions from Mods 2010 and I can't figure this one out. I think my problem is technical... We have a sample (L1,R1), ...,(Ln,Rn) with Lj and Rj normally distributed independent ...
0
votes
2answers
45 views

Probability distribution in wireless channel?

Let suppose that I have a random variable $X_{mn}=\sqrt{\left(1/d_{mn}\right)^\alpha}\times h_{mn}$ wherr $d_{mn}$ is a random variable with uniform distribution and $h_{mn}$ is a random variable with ...
1
vote
2answers
37 views

Function of random variable

I have this question: Suppose P(X=0)=1/2 and P(X=8)=1/2. What's the value of E[Y] if Y=(X^2)? So I am having trouble understanding how to go about doing this ...
0
votes
4answers
73 views

Expected value of rolling dice until getting a $3$

I am having trouble with this question with regards to random variables and calculating expected values: Suppose I keep tossing a fair six-sided dice until I roll a $3$. Let $X$ be the number of ...
0
votes
2answers
38 views

Expected value of random variable

I have this question: What's the expected value of a random variable $X$ if $P(X=1)=1/3$, $P(X=2)=1/3$, and $P(X=6)=1/3$? I am very confused as to how I can work this problem out. I was thinking ...
0
votes
2answers
77 views

Conditional expectation of number of dice rolls

I've been given the following problem and I'd like to get a better understanding of how to solve it. A fair die is rolled successively. Let $X$ be the number of rolls needed to get a 6 Let $Y$ be the ...
1
vote
2answers
35 views

Discrete random variable with $f(x)=c(2x-1)$

I feel really stupid because I don't understand this example at the start of my textbook. The chapter is discrete random variables and it starts out with this example with no explanation. I understand ...
1
vote
1answer
35 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} ...
0
votes
2answers
67 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
vote
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 ...
0
votes
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
votes
2answers
56 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 ...
0
votes
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 ...
1
vote
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? ...
0
votes
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
votes
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
votes
0answers
43 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
votes
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 ...
0
votes
0answers
23 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 ...
0
votes
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 ...
0
votes
0answers
14 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
votes
1answer
45 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
votes
1answer
129 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
votes
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
votes
0answers
85 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
vote
1answer
35 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
votes
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. ...
0
votes
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 ...
1
vote
0answers
47 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 ...
0
votes
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 ...
3
votes
0answers
52 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 ...
5
votes
2answers
65 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
votes
1answer
32 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 ?
4
votes
2answers
191 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
vote
1answer
39 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 $n$ ...
0
votes
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 ...
0
votes
1answer
41 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, ...
1
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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
votes
1answer
48 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 ...
0
votes
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
votes
1answer
28 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
votes
1answer
84 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
vote
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
votes
2answers
198 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 ...