Questions tagged [random-variables]

Question 1

Suppose we have the sum of random, independent variables $$ Z_{ij} = X_i + Y_{ij}, $$ where $X_i \sim \text{Uniform}(-d, d)$ and $Y_{ij} \sim \text{Normal}(\mu, \sigma)$. Given that only one sample of ...

Question 2

I have heard that the original proof of Chebyshev's inequality did not utilize Markov's inequality. I am interested: To know whether this is true/accurate? Any other alternative proof for Chebyshev's ...

Question 3

Recently, I read some materials on tightness of random variables and probability measures. There are two definitions: Definition 1. A sequence of random variables $(X_n)_{n \ge 1}$ is tight if for ...

Question 4

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

Question 5

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 ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 ...

Question 9

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 ...

Question 10

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 ...

Question 11

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-...

Question 12

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 ...

Question 13

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 $r_2$;...

Question 14

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 $$...

Question 15

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 ...

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Questions tagged [random-variables]

What is probability distribution function of the sum of two independent random variables when one variable is correlated with itself?

Alternative proofs for chebyshev inequality

Definition of tightness

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

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