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

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7
votes
0answers
60 views

Uncountable family of random variables

Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space $(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
6
votes
0answers
69 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
6
votes
0answers
174 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
6
votes
0answers
228 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
5
votes
0answers
171 views

expectation value for minimum distance between random variables

note: edited to clarify boundary issue Suppose $x_i$, $i=1\dots n$, are randomly drawn from a uniform circular distribution between 0 and 1 (using periodic boundaries). Let $d_i$ be the distance ...
5
votes
0answers
114 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
4
votes
0answers
74 views

Identifying the joint distribution from some values of $t \cdot X$

Suppose that $S$ is a subset of $\mathbb{R}^n$ and $X, Y$ are $\mathbb{R}^n$ valued RVs. We already know that $X$ and $Y$ are equidistributed iff $t \cdot X=^d t\cdot Y$ for all $t \in \mathbb{R}^n$. ...
4
votes
0answers
2k views

Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
3
votes
0answers
44 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
3
votes
0answers
31 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
3
votes
0answers
168 views

Skorohod representation theorem

Assume $X_n$ are random variables such that $\mathbb{P}(X_n \leq B)=1$ for some random variable $B$. Assume also $X_n \Rightarrow X$ ($X_n$ converges weakly to $X$). By the skorohod representation ...
3
votes
0answers
74 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 ...
3
votes
0answers
56 views

Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$ Is this bound (or any other form of hoeffding) tight in ...
3
votes
0answers
729 views

Relation between standard deviation and mean in random processes

In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of ...
3
votes
0answers
84 views

How many independent random variables can be defined over a standard probability space.

Consider the probability space $(0,1)$ equipped with the Borel $\sigma$-algebra and the uniform-distribution as the probability. Is there a set of independent random variables defined on it with the ...
3
votes
0answers
78 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that ...
3
votes
0answers
31 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
3
votes
0answers
26 views

How to simulate a sequence of partial sums $(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),$ given some properties.

I want to generate/simulate a sequence of partial sums. $$(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),\text{ for }1 \leq n \leq 100$$ Let $W$ be a random variable such that: $W \thicksim ...
3
votes
0answers
101 views

Random variables related through nonlinear system of equations

Lets assume two groups of random variables X and Y (the dimensionality of them is not important). I know probability distribution of X, but not of Y. I also know that Y is a function of X and they are ...
3
votes
0answers
124 views

Relation between factor graph and conditional probability distribution

First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me. The question Let say I have a factor graph illustrated in the figure. The ...
3
votes
0answers
117 views

Convergence In $L^{1}$ in the Strong Law of Large Numbers

I'm trying to prove that if $(X_n)_{n\geq 1}$ is uniformly integrable, then $X_n$ almost surely converging to $X$ implies $X_n$ converges to $X$ in $L^{1}$. How is this done? Generally speaking: ...
3
votes
0answers
325 views

On the empirical mean and variance of a Poisson i.i.d. sample

Let $X_1, X_2, \ldots, X_n$ be a random sample from a Poisson($\lambda$) distribution. Let ($\bar{X}$) be their sample mean and $s^2$ their sample variance. Show that ...
2
votes
0answers
20 views

Density of the $k^{th}$ smallest of $X_1,X_2,…,X_n$

Show that if $(X_1,X_2,...,X_n)$ are i.i.d. with common density $f$ and distribution function $F$, then $X_{(k)}$ has density $$f_{(k)}=k\binom{n}{k}f(y)(1-F(y))^{n-k}F(y)^{k-1}$$ where ...
2
votes
0answers
19 views

does the hilbert space construction of random variables allow for infinite variance?

I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner ...
2
votes
0answers
27 views

Rosanov - Probability Theory Chapter 4 Question 5

I am trying to solve one of the questions in Rosanov - Probability (Chapter 4 Question 5), but I am not exactly sure what the question is asking of me. The question is: Random variable $E$ with ...
2
votes
0answers
34 views

$\sup_nX_n<\infty$ almost surely iff $\sum_nP(X_n>A)<\infty$

Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables. Show that $\sup_nX_n<\infty$ almost surely iff there exists $A>0$ such that, $\sum_nP(X_n>A)<\infty$ By ...
2
votes
0answers
37 views

Poisson to Binomial Distribution Proof?

Q:Let {N(t) : t ≥ 0} be a Poisson process. For s = t/3, show that the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/3. Also, find the conditional distribution ...
2
votes
0answers
73 views

A measure has no point masses: is it absolutely continuous?

I have a question about measure theory. Let $\mu$ be a measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R})$. Assume that $\mu$ has no point masses - i.e. for every $a \in \mathbb{R}$, $\mu({a})=0$. Can ...
2
votes
0answers
57 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
2
votes
0answers
25 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
2
votes
0answers
69 views

A nice sequence of random variables

Let $f:U\mapsto \mathbb{R}^k$ with $U\subset \mathbb{R}$ be a smooth injective function. Suppose that $\sqrt n(Y_n- Y)\to N(0,\Omega)$ in distribution with $Y=f(X)$. Define $X_n$ by ...
2
votes
0answers
113 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 ...
2
votes
0answers
48 views

Distribution over the time it takes for a random process to reach an upper threshold

I am trying to figure out a way of determining the distribution over the time it takes for an arbitrary random process to cross a threshold value. For example, a simple (solved) case would be the ...
2
votes
0answers
29 views

Evaluate spatial variation of density-like scalar

Apologies if this has been asked previously, but I'm not totally sure of the best way to pose the question. Background I'm evaluating the variation of a spatially varying scalar field $p$ ...
2
votes
0answers
66 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
2
votes
0answers
67 views

Intuition of $ P( X = a) $ for a continuous random variable?

Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random ...
2
votes
0answers
64 views

Expectation of random variables

a) Show that $E\{X-E(X)\} = 0$ for any random variable $X$. b) Use the result in part (a) and the following equation to show that if two random variables are independent then they are uncorrelated, If ...
2
votes
0answers
63 views

When does convergence in distributions inply convergence in covariance?

Good Morning. Let $(X_n)_n$ and $(Y_n)_n$ be sequences of random variables converging in distribution respectively to $X$ e $Y$. Suppose $X_n,Y_n$ are equally distributed but dependent for all $n$, ...
2
votes
0answers
24 views

When defining the probability generating function(pgf), why do we restrict ourselves to discrete random variable?

When defining the probability generating function (pgf), why do we put the restriction of discreteness on our random variable? Replacing summation by integration, we could try to generalize it for ...
2
votes
0answers
195 views

Derive Student T distribution using transformation theorem

I am trying working on an exercise that asks me to show that If $ X_1 \in N(0,1) $ and $ X_2 \in \chi^2(n) $ are independent random variables, then $ X_1 / \sqrt{X_2/n} \in t(n) \, $ where $ ...
2
votes
0answers
122 views

Joint distribution between a uniform random variable and a function which is “almost” independent from it

Motivation Let $f(\cdot)$ be a (possibly randomized) function, such that for any random variable $X$ (taking values from a finite set $D$), $X$ and $f(X)$ are statistically independent. Let $U, U_1, ...
2
votes
0answers
95 views

The correct probability distribution / way to analyze daily changes

I am working on a report which is being sent through to end users that should flag to them any "large changes" in the day-to-day values for the past 30 days for something we would assume the ...
2
votes
0answers
92 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
2
votes
0answers
120 views

Notation for sampling a random variable

my question is pretty much the same as the one asked here: Notation for sampling random variate (I did not find a satisfying answer there...): I have a random variable $X \sim U(1, 10)$. I want to ...
2
votes
0answers
87 views

Absolute Continuity and simple discontinuity

I am reading a book called Stochastic Process, Estimation, and Control, in P.32 it states that a function with finite simple discontinuities can still be absolutely continuous, which confused me, I ...
2
votes
0answers
35 views

A fact about independent exponentially distributed RVs

Let's say $Y_i$ are independent exponentially distributed with rates $c_i$ which can be assumed to be all in $(0, \infty)$ but not necessarily the same. Whenever $t\geq0, K\geq0$ let ...
2
votes
0answers
55 views

Does this monoid have a name?

Let $S$ be the set of all real sequences $x_i$ that are equal to zero for sufficiently high indices. Consider the set of random variables $X$ with image in $S$ such that there exists an $n$ for which ...
1
vote
0answers
29 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
1
vote
0answers
36 views

An expectation inequality

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ...
1
vote
0answers
22 views

Conditional return time of simple random walk

Consider a simple random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0 = 0$. The probability to jump to the right neighbour is $p \geq \frac{1}{2}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, ...