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

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How to prove that the set of all exchangeable events is a sigma-algebra?

Let $ {X_n}_n $ be sequence of identical R.Vs Mark by S the set of all sequences available from it. An exchangeable event is $E\subset S $ which is not sensitive for finite permutations. ...
2
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1answer
21 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
0
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1answer
17 views

Terminology - “Sample space” vs “sample set”?

Given that a "sample space" is defined as the set of possible outcomes of a given random experiment, is there a fundamental reason to use the term "sample space" instead of "sample set" in probability ...
1
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1answer
23 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
1
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1answer
21 views

Expected value of function of minimum between two random variables

Two independent random variable $X,Y$ are distributed on $[0,\infty)$ according to the cumulative distribution function $F(x)=1-(x+1)^{-2}$. Let $Z=\min(X,Y)$. Determine $E\left[\frac{Z}{Z+2}\right].$ ...
2
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1answer
22 views

An Example of sequence of R.V with $E(X_n) = X_0$ but $E(X_n^{1/2}) \to 0$

I need an example of $\{X_n\}_n$ be a sequence of nonnegative, random variables, with the same finite expected value $E(X_n)=\mu_0$, that obeys: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
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0answers
9 views

Calculate best estimate of multiple mesurements with known but varying variance

When multiple experiments measure the same physical quantity and give a set of answers $s={s_1,s_2,...,s_n}$ for $n$ measurement and give an error with different variances $v={v_1,v_2,...,v_n}$. How ...
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0answers
7 views

Moments of quadratic forms

$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
1
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3answers
35 views

Definition of Random Variable on Measure Theory!

The definition is as following according to the book of John B. Walsh, Let $(\Omega, \mathbb{F}, P)$ be a probability space. A Random Variable is a real-valued function X on $\Omega$ such that for ...
0
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1answer
13 views

How to handle the noise covariance matrices in a basic Kalman Filter setup?

I've recently been trying to learn about Kalman Filters; most explanations of the Kalman Filter confuse me in what is known / unknown. I'll assume the following setup: \begin{equation} \begin{split} ...
0
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1answer
9 views

Deriving variance of a linear estimator problem

I have done parts A, B and C with no problems however part D is proving tricky: var(yi) = var(xi + vi) = var(xi) + var(vi) + 2cov(xi,vi) we know var(xi) = σ^2 and that var(vi) = w^2 and that ...
0
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0answers
7 views

Equation system with random variables

Suppose we have such system: Xt1+Ym1+Zp1+r1 = Xt2+Ym2+Zp2+r2 = Xt3+Ym3+Zp3+r3 = Xt4+Ym4+Zp4+r4 = ... (and more) where t[i], m[i], p[i] - are known variables; r[i] - are minor unknown random numbers ...
0
votes
2answers
39 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
1
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0answers
11 views

the meaning of 4-wise hash function

If someone says: 4-wise independent sign (hash) functions $s_1,s_2, s_3 : [d] → \{+1, −1\}$, then what does it means? I cannot use k-wise Independence variables (the definition 1 ...
0
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2answers
23 views

Generating points from 2 Normal distributions and $0$-probability continuous r.v.s

Consider the following experiment: We generate "green" points and "blue" points in $\mathbf{R}$ using two different normal distributions as follows: 1000 green points are sampled from a $N(-1, 1)$ ...
2
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2answers
43 views

Notation of expectation and random variables

I'm trying to understand the notation used at p18 of The Elements of Statistical Learning. I suspect errors in notation. What do the authors mean and, if any notational errors, what would be the ...
2
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1answer
29 views

Constructing dependent sequences of random variables

It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. ...
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1answer
27 views

The CDF and PDF of the transformation of a random variable (absolute value) [closed]

Let X~Exp(λ). Calculate and find the CDF and PDF of Y = |X-μ|. So far my working on paper is here, but I get stuck on how to continue. Any suggestions would be greatly appreciated! ...
0
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1answer
22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
0
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1answer
21 views

Expected value of a discrete random variable

Ok guys, I have a problem with proving this result... I have a random variable $Z$ that can take the values $[1, 2, 3]$ with probability $[\pi_1, \pi_2, \pi_3]$. How can I prove that $\mathbb{E}[Z]=2$ ...
0
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1answer
40 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
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0answers
10 views

Fitness and confidence of discrete function

New to the site, weakly educated in math, and I'm not sure if I'm stating the question in sensible terms (not even sure how to tag it), so I beg your pardon in advance: I'm receiving sequences of ...
2
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1answer
23 views

Explanation for “jointly pdf is constant but marginal pdf is not”

Consider: $X,Y \sim \text{uniformly distributed in }(0 \leq y \leq x \leq 1)$ From short computation, we know: Jointly pdf: $f_{XY}(x,y) = 2$ Marginal pdf of $x$: $f_{X}(x) =\int_0^x ...
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0answers
33 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
3
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0answers
19 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
0
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1answer
21 views

Nonlinear transform of two random variables for Gaussianity

I would like to understand the nonlinear transform of Gaussian random variable that preserves Gaussianity better when there is no $x_3$ term such that there exists a nonlinear relationship between ...
0
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0answers
42 views

Every random variable $X$ can be written as $X=\lambda Z_1+(1-\lambda)Z_2$, for $Z_1$ discrete and $Z_2$ continuous random variables.

Show that every random variable $X$ can be written as $$X=\lambda Z_1+(1-\lambda)Z_2$$ for a discrete random variable $Z_1$, a continuous random variable $Z_2$, and a real value $\lambda$. This ...
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52 views

Disjoint increments of Poisson mixture process are memory-less

Let $N(t)$ be a Poisson mixture process: $$N(t) \sim (1-p) \cdot \text{Poiss}(\lambda_0 \cdot t) \: + \: p \cdot \text{Poiss}(\lambda_1 \cdot t),$$ where $p$ is fixed and $0<p<1.$ As we ...
1
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1answer
36 views

Do 'X' and "y' have 'zero' correlation , or can be anything between '-1' and '+1'?

let , we have bi-variate data on X and Y . Now corresponding to the value $x_0$ , y can take any value.but for all other values of x , y takes a constant value. what will be the correlation ...
2
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1answer
38 views

how that $P(G)=1$ iff $\sum_n \Bbb P(A \cap E_n )=\infty$ for all events $A$ having $\Bbb P(A)>0$.

Two probability problems: 1. Let $a>0$ and let $X_n$, $n \geq 1$, be iid r.v. that are uniform on $(0,a)$ and let $Y_n = \prod_{k=1}^{n} X_k$. Determine all values of $a$ for which $\lim_{n ...
4
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1answer
49 views

Uniformly integrability and convergence

Question 1: $X_n$'s are non-negative, uniformly integrable. Then $E\left[\dfrac{\max_{1\leq k \leq n} X_k}{n}\right]\rightarrow 0$. Question 2: If u.i. is dropped then above the may fail. My ...
1
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1answer
36 views

Inequality in proof of SLLN

This comes from theorem 5.1.2 of KL Chung's A Course in Probability Theory. Suppose ${X_n}$ are uncorrelated and their second moments have a common bound. Then For each $n \ge 1 $, $D_n:= ...
3
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1answer
50 views

$\limsup \frac{|S_n|}{n}=\infty$

$X_n$'s are i.i.d symmetric with $E|X_1|=\infty$. Then $\limsup \frac{|S_n|}{n}=\infty$. How do I show $\limsup \frac{S_n}{n}=\infty$ and $\liminf \frac{S_n}{n}=-\infty$? My attempt: Let $c=\limsup ...
0
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0answers
29 views

How is $\mathcal F_\infty$ different from $\bigcup_{n=0}^\infty \mathcal F_n$? [duplicate]

Let $(X_n)_n$ be a sequence of random variables. Define $\mathcal F_\infty := \sigma(X_0, X_1, \ldots)$ and $\mathcal F_n := \sigma(X_0, X_1, \ldots, X_n)$. In the proof of the Kolmogorov's zero–one ...
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2answers
31 views

How to model the probability of a system failing if any of its subsystems fails?

Suppose I have a system that consists of $m$ independent subsystems. The work expectancy for each of these subsystems is a random variable, let's say $X_i$, where $i$ denotes the $i$-th subsystem. ...
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0answers
31 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
4
votes
1answer
59 views

iff $E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$

Question: $X_n$'s are i.i.d then $$E\Bigl(\sup_{n\geq 1} \frac{|X_n|}{n}\Bigr)<\infty \iff E\Bigl(|X_1|\log ({1+|X_1|)}\Bigr)<\infty$$ My attempt: for $\Rightarrow$ part, because $\limsup ...
4
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2answers
228 views

Is “random variable” really random?

This is a concept question. The fundamental of modern probability theory is measure theory. A probability space is just a finite measure space and a random variable is just a measurable function. We ...
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3answers
74 views

Show that the random variables $Z = XY \sim N(0,\alpha^2)$.

The random variables $X$ and $Y$ are independent with $$ f_X (x) = x/α^2 e^{-x^2/2α^2} \text{ for } x > 0 $$ and $$ f_Y(y) = \frac {1} {\pi \sqrt{1-y^2}} \text{ for } |y|<1 $$ and is zero ...
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0answers
24 views

Probability of Minimum of random set

I have a probability confusion on a queueing problem. Let arrival rate $\lambda_k$ = $p_k\lambda_k + (1-p_k)\lambda_k$, how can we find the minimum of the random variables, say by $A_k$ with arrival ...
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2answers
30 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
0
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1answer
30 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
0
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0answers
28 views

Exponential Probability - Light bulb lifetime question

Suppose the lifetime of a cheap light bulb is exponentially distributed with average lifetime equal to 100 hours. If you have a supply of 100 such bulbs and you use one at a time, with a failed one ...
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0answers
26 views

Linear combination of i.i.d random variables

We say that a random variable $X$ satisfies the $(\alpha,\beta)-$condition for some $\alpha>0$ and $\beta>0$ if $$\mathbb{P}(|X|<t)<\alpha t\text{ and }\mathbb{P}(|X|>t)<e^{-\beta ...
0
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1answer
29 views

MI between vectors

There exist a relationship between MI and entropy of two random variables i-e $$ I(X;Y)=H(Y)-H(Y|X).$$ But what if $ \overrightarrow X \: \in \mathbb {\{0,1\}}^2$ and $ \overrightarrow Y \: \in ...
2
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1answer
76 views

Packing of discrete random variables with finite second moment

I am considering a discrete random variable $X \in\mathbb{R}$ with $N$ points (where each point has non-zero probability) and $E[X^2]=1$ and $E[X]=0$. Let $d_l$ be the the smallest distance between ...
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1answer
29 views

Is there something wrong with this Probability Density Function?

Problem: A certain software company uses a certain software to check for errors on any of the programs it builds and then discards the software if the errors found exceed a certain number. Given that ...
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0answers
25 views

Law of large numbers for arbitrarily dependent random variables

Consider a sequence of random variables $X_1,X_2,...,X_n$. No assumptions abou independence is made. Only joint probability density function is known, i.e. $f(x_1,...,x_n)$. Then Markov's theorem ...
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1answer
66 views

Questions on finding expected value and variance on a Poisson distribution

FProblem: A student walks along a real line and tries to get to the origin. Each step he makes is random ; the larger the intended step, the greater the variance is of that step. When the student is ...
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2answers
37 views

Conditional variance of sum of two correlated random variables

Let $\theta\sim\mathcal{N}(\bar{\theta},1/\tau_\theta)$ be a normally distributed random variable, $\varepsilon\sim\mathcal{N}(0,1/\tau_\varepsilon)$ be a normally distributed noise term independent ...