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

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0
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0answers
9 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
4
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6answers
68 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) ...
3
votes
2answers
101 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
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1answer
22 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + ...
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0answers
36 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
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0answers
25 views

Conditional expectation of discrete uniform random variables with one fixed

Came across a problem that I worked on sometime ago having the following structure: Given an opaque container (or locomotive with so many passenger cars, etc) that has--with equal probability--1 to N ...
0
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1answer
28 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
0
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1answer
42 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
0
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1answer
37 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
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2answers
64 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
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2answers
58 views

Random increment through a probability distribution function

To Clarify i am trying to generate a random variable from a gamma pdf If $\Delta X$ indicates a random increment and it is said that $\Delta X$ follows a Gamma distribution. What would that mean ...
0
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1answer
54 views

For zero-mean r.v. $X$ with var. $\sigma^2$, want to show $E[e^{2X}]\leq e^{2\sigma^2}$.

Let $X$ have zero mean, $E[X]=0$ and finite variance $E[X^2]=\sigma^2<\infty$. I'm trying to show $$ E[e^{2X}] \leq e^{2\sigma^2}. $$ I started out with this related question, but I hadn't quite ...
0
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1answer
28 views

Convergence in proability does not imply convergence a.s. [duplicate]

I know convergence in probability does not imply convergence in measure. I would like to see some simple example. Do you have any ideas please?
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2answers
41 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
0
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0answers
14 views

Generating pseudo-random numbers around a distribution with an uncertain/chaotic mean

I originally asked this question on cross-validated, but apparently it is too mathematical a problem for that site. I want to simulate data collected by an instrument realistically. The problem is ...
0
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2answers
17 views

statistics question about two random variables

Two random variables $X$ and $Y$ have the following joint pdf: $$f(x, y) = \begin{cases} \frac35x(y + y^2) & \text{if }0<x<2\text{ and } 0 < y < 1\\ 0 & \text{otherwise} ...
1
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0answers
26 views

mean and variance of this Gaussian random variable

I am trying to read through this paper - http://www.malcolmdshuster.com/Pub_2002c_J_scale_scan.pdf Equation 2(b)from the paper says [A] $\nu_k \equiv 2(B_k - b).\epsilon_k - |\epsilon_k|^2 $ where ...
1
vote
2answers
55 views

Bounding the expectation of a function of a zero-mean random variable

I have a random variable $X$ with mean zero, $E[X]=0$, and finite second moment, $E[X^2]=\sigma^2<\infty$. I'm wondering if it's possible to show the following bound: $$ E[(e^{X/2}-1)^2] \leq ...
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1answer
90 views

How can I prove if $Y\leq X$ then $E[Y]\leq E[X]$?

If $Y\leq X$ always holds, then $E[Y]\leq E[X]$. How can I prove this (formally)? Also, can the equality happen if we know that $Y=X$ does not always hold? (i.e. $X$ and $Y$ are not exactly the same)
0
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1answer
25 views

Check if the weak law of large numbers holds true for the following sequence of random variables

Suppose we have $n$ independent discrete random variables, whose distribution is as follows: $X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values: $-\sqrt{k}$ with a ...
0
votes
1answer
67 views

Probability distribution of bored people

5 people are arranged in a row, a person is talkative with a probability of $p$ and silent with a probability of $1-p$, each is independent. A person is bored if he's talkative and sits between two ...
1
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0answers
33 views

Expectation of $\min(X, c)$ for $X$ truncated r.v. and $c$ constant

I have a random variable $X$ and a constant $c\geq 0$. I define the r.v. $Y = \min(X, c)$ and I want to calculate $E[Y]$. I have seen different posts on similar topics, so I am trying to pull all ...
1
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1answer
35 views

What is the probability density function of **the multiplication of Gaussian variables**?

Assuming $x_1,x_2,\ldots, x_n$ are $n$ independent variables from standard Gaussian distribution $N(0,1)$. Then we construct a new variable by $y=\Pi_{i=1}^n x_i$. Can anyone show the probability ...
0
votes
3answers
104 views

How is it possible for two random variables to have same distribution function but not same probability for every event?

It is completely out of the world for me to hear that such a case exists. I was shocked and could not develop any intuition as to how it is possible. It also breaks my understanding (intuitive) of the ...
1
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0answers
41 views

Defining a function over time in terms of a random variable that is undefined at a certain time

Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$. Also, for each $n \in \mathbb{N}$, let $F_n$ be ...
1
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0answers
24 views

How to decorrelate/Whiten a non-white additive random variable?

I have a signal processing problem where I have the Additive Noise Model (assume Gaussian noise). $$ y = x + w $$ where, $y$ is corrupted signal, $x$ is original signal & $w$ is a non-white ...
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votes
1answer
86 views

what is the probabilty that sum of two random numbers between A and B is less than third number C [closed]

What is the probabilty that sum of two random numbers uniformly distributed in $[A,B]$ is less than a fixed $C$? I have tried answering this question using graph method to find the area under the ...
1
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0answers
32 views

How to prove: $E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$?

How to prove: $$E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$$, where $k$ is a positive integer, $x,y$ are fixed vectors, each entry in $w$ i.i.d. follows from an standard norm ...
1
vote
1answer
35 views

About 'Marcinkiewicz–Zygmund inequality'

Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki ...
1
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0answers
23 views

A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...
3
votes
1answer
155 views

CLT for independent, but non-identically distributed exponential variables

This problem is practice for my qualifying exam and comes from Resnick, chapter 9. Could anyone comment on my solution(s)? Problem Suppose ${e_n, n\ge 1}$ are independent exponentially distributed ...
3
votes
3answers
73 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
0
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1answer
47 views

Definition of independence of infinite random variables

When random variables $Y_1, Y_2, ... Y_n$ are independent, we say that $$P\left(\bigcap_{i=1}^{n} (Y_i \in B_i)\right) = \prod_{i=1}^{n} P(Y_i \in B_i)\tag{F1}$$ or for any distinct indices $i_1, ...
2
votes
0answers
30 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
0
votes
1answer
30 views

What is the distribution of the sum of several normally distributed random variables?

Let's say we have n normally distributed random variables all with the same median and variance. Do we have a possibility to estimate the distribution law of the sum of those variables? I assume ...
1
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1answer
56 views

What is $E[X|Y]$ if the random variables $X$ and $Y$ are independent?

Just looking for an explanation of how the conditional expectation "$E[X|Y]$" of any two random variables $X$ and $Y$ would change if we included the condition that $X$ and $Y$ be independent. ...
1
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1answer
36 views

Probability of inequality with Chi-squared distributed random variables

I want to evaluate analytically the following probability: $P(X+XY\leq Z+ZW)$ where $X\thicksim \chi_1^2$, $Y\thicksim \chi_a^2$, $Z\thicksim \chi_1^2$, and $W\thicksim \chi_b^2$ with $a,b\geq 2$. ...
1
vote
2answers
28 views

Find the expected frequency of some state in a state sequence of length N given a transition matrix M

I can represent stochastically-articulated sequences of states using a transition matrix M where a given entry in cell (i,j) corresponds to the probability of state j given that the current (or, most ...
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votes
5answers
48 views

PDF of function of uniform random variable [closed]

Why PDF of $g(X)=X^3$ is not uniformly distributed, when X is uniform random variable between $(0,1)$? As for every value of X there is unique value of $g(X)$, hence the probability density of $g(X)$ ...
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1answer
51 views

Existence of a global maximum of a function defined with the moment-generating function

Can someone give me an idea how to prove the following exercise? Let $Z$ be a real-valued random variable whose moment-generating function $m_Z$, with $m_Z(\gamma)= E\left[ \exp(\gamma Z) \right]$, ...
2
votes
0answers
59 views

If $x_n\in S$ and $x_n\to x$ then $x\in S$

Suppose $S$ is the support of a univariate cdf $F$. If $x_n\in S$ is a sequence of reals such that $x_n\to x$ then show that $x\in S$. I believe the question is actually very very simple, in the ...
3
votes
1answer
70 views

When do we say independence is the probability of intersection being equal to the product of probabilities? [duplicate]

This is something I never really got in either Elementary Probability Theory or Advanced Probability Theory because my professors mainly discussed independence between 2 objects. Please tell me if my ...
0
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0answers
14 views

What are some error measures used for fitting PMFs?

I have a given PMF, $f_X(x)$, and am trying to create a fitted PMF, $g_X(x)$, that comes "as close as possible" to it, but am not sure what to use as a measure of fit. Simply minimizing standard error ...
1
vote
1answer
17 views

How to prove **$k_1$-wise independence implies $k_2$-wise independence if $k_1 \geq k_2$**

The Definition 1 shows the the meaning of k-wise independence. So can anybody help to prove $k_1$-wise independence implies $k_2$-wise independence if $k_1 \geq k_2$? By the way, I did not ...
0
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0answers
14 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
1
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0answers
33 views

$E(x\mid xy)$ linear in $xy$

I read a paper that makes an assumption that $E(x\mid xy)$ linear in $xy$ where $x$ and $y$ are independent. I was wondering what is fundamental about this assumption, or in other words, what ...
0
votes
1answer
19 views

Equivalence of the second moment of two random variables when their first moments and covariance with a third random variable are equal

I'm trying to check under which conditions the standard deviation of two random variables is identical when I know some properties about other moments of these random variables. I suppose that their ...
0
votes
1answer
34 views

Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
1
vote
1answer
26 views

Using Gronwall's Inequality with Random Variables

Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form: $$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$ ...
1
vote
1answer
65 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...