Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
12,566 questions
2 votes
1 answer
46 views
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 ...
- 1,019
3 votes
1 answer
52 views
Conditional probability for linear combinations of independent exponentials
I am working on the following exercise. Let $$X_1 \sim \mathrm{Exp}\left(\tfrac12\right), \qquad X_2 \sim \mathrm{Exp}\left(\tfrac12\right),$$ independent. Define $$Y_1 = X_1 + 2X_2, \qquad Y_2 = 2X_1 ...
- 375
0 votes
0 answers
49 views
Characterization of distribution preserving automorphisms [closed]
Let $X$ be a random variable $(\Omega, \mathcal{A}, P) \to (\mathbb{R}, \mathcal{B(R)} ) $, I want to know if we can characterize the maps in $\mathcal{Aut}(\Omega)$ that fix the distribution of $X$. ...
- 14.5k
2 votes
2 answers
118 views
Intuitive Explanation for Convergence in Probability and Convergence in Distribution
Having a bit of trouble with the definitions for convergence in probability and convergence in distribution for random variables. The textbook (Degroot) defines each as follows: Convergence in ...
- 21
0 votes
1 answer
63 views
How is the normalization property(i.e.the sum equals 1)of the joint probability mass function for a two-dimensional discrete random variable ensured?
In the following,we assume that two-dimensional discrete random variables $\vec{X}=[X_1,X_2]$ on $\mathbb{R} ^2$,and the range of values for both $X1$ and $X2$ is countably infinite,and they are ...
- 161
5 votes
1 answer
284 views
Question regarding Jensen's inequality when it comes to logarithm
Let $X$ be a real-valued random variable, and define its moment generating function (MGF) as $$ M_X(s) = \mathbb{E}[e^{sX}], $$ where $\mathbb{E}[\cdot]$ denotes the expected value of the random ...
- 191
3 votes
1 answer
87 views
Derivation of Diffusion Equation in 1-D
I am trying to rigorously derive the diffusion equation, given by $$ \frac{\partial u}{\partial t} = D\,\frac{\partial^2 u}{\partial x^2}, \qquad D = \frac{h^2}{2\tau}. $$ from a simple one-...
- 3,585
0 votes
0 answers
55 views
Can we have a random variable with mixed joint distribution resulting in a singular and a non-singular marginal distribution?
This question may be a little trivial, but I was wondering if we can construct a bivariate (or multivariate) probability distribution function in a way that we have a mix of a singular and an ...
- 51
6 votes
1 answer
617 views
Median wealth after repeated iterations of multiplicative game?
I start with \$1. After one iteration of a game, one of the following $m$ outcomes occurs: With probability $p_1$, my wealth multiplies by $r_1$; With probability $p_2$, my wealth multiplies by $r_2$;...
- 163
0 votes
1 answer
60 views
Does decomposition of PDFs guarantee independence of random variables?
Is this conjecture correct? If not, can it be modified to a correct one: Let $X,Y$ be continuous RVs with joint PDF $f(x,y)$. Then $X,Y$ are independent iff there exists functions $g, h$ such that $$...
- 6,361
1 vote
1 answer
95 views
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 ...
- 399
1 vote
0 answers
50 views
$\sigma(f(X))\subseteq \sigma(X)$ 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}, \...
- 742
1 vote
0 answers
61 views
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 ...
- 11
2 votes
1 answer
158 views
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 $2$...
- 219
3 votes
1 answer
109 views
Connections between similar looking theorems
Consider the following three theorems: If $m,n$ are relatively prime, then $\varphi(mn) = \varphi(m)\varphi(n)$ (Where $\varphi$ is the totient function, the Euler-phi function) If $f: A \rightarrow ...
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