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

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1
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1answer
20 views

Two uncorrelated random variables both taking only two values are independent

Let X and Y be random variables both taking only two values. Show that if they are uncorrelated then they are independent.
-2
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1answer
10 views

Question about joint convergence with covariance

Just a short question about joint convergence. Assume random variables $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ and the following convergences in distribution $X_n \to X$, $Y_n \to Y$. Furthermore, ...
-3
votes
0answers
25 views

a question about sample space [on hold]

How to represent the following statement mathematically: "The event $\{A_n \text{occurs infinitely often} \}$ is $\{ \omega \in \Omega | \omega \in A_n \ \text{for infinitely many values of} \ n ...
0
votes
1answer
20 views

Monotone likelihood property and first order stochastic dominance

I have a question regarding first order stochastic dominance. Give two pdf $f(x)$, $g(x)$, $x\in[x_0,x_1]$. For all $x$ on the support, I have $$ g(x) = f(x)\cdot H(x) $$ where $H(x)$ is continuous, ...
0
votes
1answer
47 views

Generation of random variable from a complicated CDF

Suppose I am given a CDF of a distribution, given by $F(x) ∝ \int_0^1 x^y e^{-y} dy.$ Here,'x' ranges from 0 to 1. How do I generate a random variable from this distribution?
0
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0answers
12 views

Uniformly distributed points in a spheroid

I want to pick points inside a spheroid with uniform probability. The problem was solved for a sphere here: Uniformly distributed points on a sphere. I know, that the "shooting method" works: Generate ...
0
votes
0answers
20 views

How to model time changing random variables

Lets say I have a random variable $X(t)$ which describes some unit of motion of a living organism and $X(t)$ is itself a timeseries since this unit of motion changes in time. I would like to be able ...
0
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0answers
30 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
0
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0answers
32 views
+50

Consequences of exchangeability of random variables

Consider two random variables $X_i$ and $U_i$ respectively distributed as $F_{X_i}(\cdot)$ and $F_{U_i}(\cdot)$ for $i=1,...,N$. Let $X:=(X_1,...,X_N)$ and $U:=(U_1,...,U_N)$ be respectively ...
1
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0answers
14 views

PMF of discrete conditional random variable

Let $X$ be a discrete random variable (r.v) whose range is the set of non-negative integers. Let the probability mass function (PMF) of $X$ be: $PX(i)=P[X=i]=kp^i, s.t. i = 0, 1, 2, ...$ where $p \in ...
2
votes
1answer
43 views

transformation of uniformly distributed random variable f(x)=1/2pi into Y=cosx

Let $X$ be a uniformly distributed function over $[-\pi􀀀;\pi]$. That is $ f(x)=\left\{\begin{matrix} \frac{1}{2 \pi} & -\pi\leq x\leq \pi \\ 0 & otherwise \end{matrix}\right.\\ $ Find ...
0
votes
0answers
16 views

Variance of estimating coefficients by correlating a sequence

I have a sequence $$ r[n] = a_1.t_1[n] + a_2.t_2[n] + a_3.t_3[n] + ... $$ where $t_1, t_2, t_3,...$ are uncorrelated, two-level (+A/-A), zero mean, pseudo-random sequences. To estimate $a_1$, ...
-2
votes
1answer
96 views

Conditional distribution of mixed process

Let $Y$ be a random variable such that: $$Y \sim \begin{cases} \operatorname{Poiss}(\lambda_0), & x= 0 \\ \operatorname{Poiss}(\lambda_1), & x=1 \\ \end{cases} $$ where ...
1
vote
1answer
25 views

Change of variable using dirac delta function

How do I intuitively understand the following result to find the probability density function $P_Y(y)$ given $P_X(x)$ after change of variables $y=f(x)$ or several variables. How to derive this from ...
0
votes
2answers
28 views

Name of the probability distribution

If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows: $$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$ ...
0
votes
1answer
22 views

Expectation of the maximum of n random variables?

Let's say we have $n$ independent random variables, each variable equally likely to take any value in the interval $[0,1]$. What is the expectation of the maximum of these $n$ random variables? ...
1
vote
1answer
34 views

Writing the expected value of a random variable in terms of its cumulative distribution function

My professor said that an alternative expression for the expected value of a random variable can be written as: $$ E[X] = \int_{0}^{\infty} (1-F_X(x)) \, dx - \int_{-\infty}^0 F_X(x) \, dx $$ No ...
0
votes
1answer
28 views

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
votes
1answer
29 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
votes
1answer
21 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
31 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
vote
1answer
24 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
votes
1answer
23 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: $E(\sqrt{X_n})>E(\sqrt{X_{n+1}})>\dots \to 0$
1
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0answers
10 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 ...
0
votes
0answers
8 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
vote
3answers
36 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
votes
1answer
17 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
votes
1answer
10 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
votes
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
40 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
vote
0answers
12 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
votes
2answers
24 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
votes
2answers
44 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
votes
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 $X$. ...
-1
votes
1answer
31 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
votes
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
votes
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
votes
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, ...
0
votes
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
votes
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 ...
0
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0answers
35 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
votes
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
votes
1answer
23 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
votes
0answers
43 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 ...
1
vote
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
votes
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
votes
1answer
50 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
vote
1answer
38 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
votes
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
votes
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 ...