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

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3
votes
0answers
42 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
3
votes
1answer
35 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
0
votes
0answers
19 views

Seeking for an example of Bayesian estimation of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayesian estimation of one unknown parameter of random variables (either ...
0
votes
0answers
13 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
0
votes
1answer
29 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
1
vote
0answers
9 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. Let its CDF be given by $F(\theta) := ...
2
votes
2answers
400 views

Probability of inequality between random variables

In order to prove a theorem in my research, I would like to use a lemma on basic probability theory, but I don't know if it is correct. For three random variables $X,Y$, and $Z$ not necessarily ...
1
vote
1answer
24 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
0
votes
1answer
37 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
1
vote
1answer
39 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
0
votes
0answers
23 views

Is it possible to exchange a sum in a conditional expectation

Let $X_1, X_2, \ldots \geqslant 0$ and $Y$ be RVs over $\mathbb{R}^n$. Then is it true that $\mathbf{E} \left[ \sum_{i = 1}^{\infty} X_i \mid Y \right] = \sum_{i = 1}^{\infty} \mathbf{E} [X_i \mid ...
0
votes
2answers
49 views

Prove (or disprove) that $\mathbb{E}[X]\geq 0$ for positive random variable.

Let $X$ be a random variable such that $X\in[0,1]$. I was wondering if $\mathbb{E}[X]$ must be $\geq0$. Since $X$ is a positive random variable, we can apply the Markov-inequality: for each positive ...
0
votes
2answers
47 views

what is the expected value of $x^TAx$? [on hold]

Assume $x\in \mathbb{R}^N$ is a random variable vector (like a noise sequence). You now want to calculate the following term: $E\{x^{T}Ax\}$, where $A$ is a constant matrix. How can this expression ...
0
votes
2answers
34 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [on hold]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
0
votes
0answers
8 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 = ...
-3
votes
1answer
21 views

Uniform distribution and real values [on hold]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
4
votes
6answers
61 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
89 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 ...
0
votes
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) + ...
0
votes
0answers
27 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, ...
0
votes
0answers
20 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
votes
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
votes
0answers
19 views

Probability distribution to failure [closed]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...
0
votes
1answer
32 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
votes
1answer
32 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 $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
0
votes
2answers
58 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, ...
0
votes
2answers
56 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
votes
1answer
52 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
votes
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?
0
votes
1answer
27 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
votes
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
votes
2answers
16 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
vote
0answers
23 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
53 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 ...
-1
votes
1answer
86 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
votes
1answer
23 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
vote
0answers
32 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
vote
1answer
32 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
97 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
vote
0answers
40 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
vote
0answers
21 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 ...
-4
votes
1answer
68 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
vote
0answers
31 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
34 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
vote
0answers
22 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 $N$ ...
-1
votes
0answers
35 views

With the constraint $E(X)=0,E(X^2)=1$, is Rademacher (symmetric Bernoulli) variable X the best choice to minimize $E(X^4)$?

Rademacher variable $X$ means that $X$ can be either $-1$ or $1$ with equal probability $0.5$. Then my question is that: Is Rademacher (i.e. symmetric Bernoulli) variable X the best choice to ...
-2
votes
0answers
43 views

calculate $E(X^2),E(X^4),\ldots$ for various random variables

Is there any document or tool directly showing the results of $E(X^2),E(X^4),\ldots$ for various random variables? where $E$ is the expectation and $X$ is a kind of random variable that may follow ...
3
votes
1answer
148 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
71 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: ...