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Question 1

2votes

2answers

88views

Blind Spot Regarding the Correlation Coefficient

I've noticed that there is a strange unexplained thing about Pearson correlation coefficient$$\rho(X,Y) = \frac{\operatorname{Cov} (X,Y)}{\sqrt {\operatorname{Var}X} \sqrt {\operatorname{Var}Y} }$$...

Alex

153

asked12 hours ago

Question 2

2votes

1answer

41views

complete measurable space with respect to pushforward measure

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $(\mathcal{X}, \mathcal{F})$ be a measurable space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ a random ...

guest1

766

askedyesterday

Question 3

6votes

4answers

383views

Expectation of an absolute value

Let $X,Y$ be two i.i.d.I am trying to bound the expectation of how afar from one another they can get?That is, $E[|X-Y|]$. I know that:$$ E[X-Y] = E[X]- E[Y] = 0$$But what about $|X-Y|$?One ...

Eric_

1,059

askedNov 21 at 13:08

Question 4

3votes

1answer

70views

Conditional probability for linear combinations of independent exponentials

I am working on the following exercise.Let$$X_1 \sim \mathrm{Exp}\left(\tfrac12\right), \qquadX_2 \sim \mathrm{Exp}\left(\tfrac12\right),$$independent. Define$$Y_1 = X_1 + 2X_2, \qquad Y_2 = 2X_1 ...

Pizza

377

askedNov 20 at 15:37

Question 5

2votes

2answers

125views

Intuitive Explanation for Convergence in Probability and Convergence in Distribution

Having a bit of trouble with the definitions for convergence in probability and convergence in distribution for random variables. The textbook (Degroot) defines each as follows:Convergence in ...

itsmarisa

21

askedNov 9 at 9:22

Question 6

0votes

1answer

68views

How is the normalization property(i.e.the sum equals 1)of the joint probability mass function for a two-dimensional discrete random variable ensured?

In the following,we assume that two-dimensional discrete random variables $\vec{X}=[X_1,X_2]$ on $\mathbb{R} ^2$,and the range of values for both $X1$ and $X2$ is countably infinite,and they are ...

user1405622

161

askedNov 8 at 5:09

Question 7

5votes

1answer

290views

Question regarding Jensen's inequality when it comes to logarithm

Let $X$ be a real-valued random variable, and define its moment generating function (MGF) as$$M_X(s) = \mathbb{E}[e^{sX}],$$where $\mathbb{E}[\cdot]$ denotes the expected value of the random ...

Anne

191

askedNov 4 at 21:28

Question 8

3votes

1answer

106views

Derivation of Diffusion Equation in 1-D

I am trying to rigorously derive the diffusion equation, given by$$\frac{\partial u}{\partial t} = D\,\frac{\partial^2 u}{\partial x^2},\qquadD = \frac{h^2}{2\tau}.$$from a simple one-...

sam wolfe

3,585

askedNov 4 at 17:57

Question 9

0votes

0answers

55views

Can we have a random variable with mixed joint distribution resulting in a singular and a non-singular marginal distribution?

This question may be a little trivial, but I was wondering if we can construct a bivariate (or multivariate) probability distribution function in a way that we have a mix of a singular and an ...

Lucas

41

askedNov 3 at 12:54

Question 10

6votes

1answer

622views

Median wealth after repeated iterations of multiplicative game?

I start with \$1. After one iteration of a game, one of the following $m$ outcomes occurs:With probability $p_1$, my wealth multiplies by $r_1$;With probability $p_2$, my wealth multiplies by $r_2$;...

Andrés Mejía

163

askedOct 26 at 21:08

Question 11

0votes

1answer

64views

Does decomposition of PDFs guarantee independence of random variables?

Is this conjecture correct? If not, can it be modified to a correct one:Let $X,Y$ be continuous RVs with joint PDF $f(x,y)$. Then $X,Y$ are independent iff there exists functions $g, h$ such that $$...

SRobertJames

6,461

askedOct 26 at 12:29

Question 12

1vote

1answer

108views

Commutant of random linear combination of matrices

I'm not too familiar with random matrix theory so I cannot find a suitable reference for this question.Consider a set of matrices $\{A_i\}_{i=1}^k\subseteq M_{d\times d}$ over the complex field and ...

Another User

399

askedOct 24 at 17:23

Question 13

1vote

0answers

52views

$\sigma(f(X))\subseteq \sigma(X)$ and $\sigma(f(X))=\sigma(X)$ if and only if $f$ is an injective or bijective map?

Let $(\Omega, \mathcal{A})$ be a measurable space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ a measurable function and $f:(\mathcal{X}, \mathcal{F})\rightarrow (\mathcal{Z}, \...

guest1

766

askedOct 24 at 11:12

Question 14

1vote

0answers

61views

Continuous-time bounded stochastic processes: Do they take values arbitrarily close to the bound in non-zero intervals?

As background, I am an academic working in engineering with quite some maths experience. However, my experience in probability theory for continuous-time processes is limited.Let's say we have a ...

user1704396

11

askedOct 21 at 13:35

Question 15

2votes

1answer

167views

How to break the cases when we have more than two random variables involved with algebra of modulus quantities?

I have two (two-part) questions from the context of transformation of random variables. I specifically want to understand how the breaking of cases work for such following problems, where more than $2$...

TopoSet32

219

askedOct 17 at 16:53

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Stack Exchange Network

Questions tagged [random-variables]

Blind Spot Regarding the Correlation Coefficient

complete measurable space with respect to pushforward measure

Expectation of an absolute value

Conditional probability for linear combinations of independent exponentials

Intuitive Explanation for Convergence in Probability and Convergence in Distribution

How is the normalization property(i.e.the sum equals 1)of the joint probability mass function for a two-dimensional discrete random variable ensured?

Question regarding Jensen's inequality when it comes to logarithm

Derivation of Diffusion Equation in 1-D

Can we have a random variable with mixed joint distribution resulting in a singular and a non-singular marginal distribution?

Median wealth after repeated iterations of multiplicative game?

Does decomposition of PDFs guarantee independence of random variables?

Commutant of random linear combination of matrices

$\sigma(f(X))\subseteq \sigma(X)$ and $\sigma(f(X))=\sigma(X)$ if and only if $f$ is an injective or bijective map?

Continuous-time bounded stochastic processes: Do they take values arbitrarily close to the bound in non-zero intervals?

How to break the cases when we have more than two random variables involved with algebra of modulus quantities?

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